How do the Catholic and LDS views of what a prophet is differ? Answers can address either denomination or both. My most recent understanding on the subject was that a prophet loses awareness (a trance) and then receives the word or message of God. However, I don't know if these are the criteria per se, or if they are part of an overarching theme. (My follow-up question will be, what do we call people like King David or the sons of Korach - but please don't answer that here.)
How do the Catholic and LDS views of what a prophet is differ?
First of all, let us deal with what a prophet is within Catholic understanding.
Catholic generally only apply the title of prophet to those individuals named in Sacred Scriptures. Many Catholic Blesseds and Saints have received the genuine gift of prophecy as been given by the Holy Spirit, but the Church does not call them prophets. The word Prophet is traditional reserved for the Biblical Prophets of Old.

Prophet
Definition
The biblical term &quot;nabi&quot; means one who spoke, acted, or wrote under the extraordinary influence of God to make known the divine counsels and will. Yet commonly associated with this primary function to proclaim the word of God, a prophet also prophesied by foretelling future events. His role, then, was to both proclaim and to make the proclamation credible.

The Catholic Encyclopedia in its’ article entitled Prophecy, Prophet, and Prophetess, goes into much greater detail on the Catholic viewpoint as to what is a Prophet.

General idea and the Hebrew names
(1) General Idea — The Hebrew Prophet was not merely, as the word commonly implies, a man enlightened by God to foretell events; he was the interpreter and supernaturally enlightened herald sent by Yahweh to communicate His will and designs to Israel. His mission consisted in preaching as well as in foretelling. He had to maintain and develop the knowledge of the Old Law among the Chosen People, lead them back when they strayed, and gradually prepare the way for the new kingdom of God, which the messias was to establish on earth. Prophecy, in general, signifies the supernatural message of the Prophet, and more especially, from custom, the predictive element of the prophetic message.
(2) The Hebrew Names — The ordinary Hebrew for prophet is nabî'. Its etymology is uncertain. According to many recent critics, the root nabî, not employed in Hebrew, signified to speak enthusiastically, &quot;to utter cries, and make more or less wild gestures&quot;, like the pagan mantics. Judging from a comparative examination of the cognate words in Hebrew and the other Semitic tongues, it is at least equally probable that the original meaning was merely: to speak, to utter words (cf. Laur, &quot;Die Prophetennamen des A.T.&quot;, Fribourg, 1903, 14-38). The historic meaning of nabî' established by biblical usage is &quot;interpreter and mouthpiece of God&quot;. This is forcibly illustrated by the passage, where Moses, excusing himself from speaking to Pharao on account of his embarrassment of speech, was answered by Yahweh: &quot;Behold I have appointed thee the God of Pharao: and Aaron thy brother shall be thy prophet. Thou shalt speak to him all that I command thee; and he shall speak to Pharao, that he let the children of Israel go out of his land&quot; (Exodus 7:1-2). Moses plays towards the King of Egypt the role of God, inspiring what is to be uttered, and Aaron is the prophet, his mouthpiece, transmitting the inspired message he shall receive. The Greek prophetes (from pro-phanai, to speak for, or in the name of someone) translates the Hebrew accurately. The Greek prophet was the revealer of the future, and the interpreter of divine things, especially of the obscure oracles of the pythoness. Poets were the prophets of the muses: Inspire me, muse, thy prophet I shall be&quot; (Pindar, Bergk, Fragm. 127).
The word nabî' expresses more especially a function. The two most usual synonyms ro'éeh and hozéh emphasize more clearly the special source of the prophetic knowledge, the vision, that is, the Divine revelation or inspiration. Both have almost the same meaning; hozéh is employed, however, much more frequently in poetical language and almost always in connexion with a supernatural vision, whereas râ'ah, of which ro'éh is the participle, is the usual word for to see in any manner.
Brief sketch of the history of prophecy
(1) The first person entitled nabî' in the Old Testament is Abraham, father of the elect, the friend of God, favoured with his personal communications (Genesis 20:7). The next is Moses, the founder and lawgiver of the theocratic nation, the mediator of the Old Covenant holding a degree of authority unequalled till the coming of Jesus Christ. &quot;And there arose no more a prophet in Israel like unto Moses, whom the Lord knew face to face, in all the signs and wonders, which he sent by him, to do in the land of Egypt to Pharao, and to all his servants, and to his whole land, and all the mighty hand, and great miracles, which Moses did before all Israel&quot; (Deuteronomy 34:10 sqq.). There were other Prophets with him, but only of the second rank, such as Aaron and Maria, Eldad and Medad, to whom Yahweh manifested himself in dreams and vision, but not in the audible voice with which He favoured him, who was most faithful in all His house (Numbers 12:7).
Of the four institutions concerning which Moses enacted laws according to Deuteronomy (14:18-18), one was prophecy (18:9-22; cf. 13:1-5, and Exodus 4:1 sqq.). Israel was to listen to the true Prophets, and not to heed the false but rather to extirpate them, even had they the appearance of miracle-workers. The former would speak in the name of Yahweh, the one God; and foretell things that would be accomplished or be confirmed by miracles. The latter were to come in the name of the false gods, or teach a doctrine evidently erroneous, or vainly endeavour to foretell events. Later prophetic writers added as other signs of the false Prophets, cupidity, flattery of the people or the nobles, or the promise of Divine favour for the nation weighed down with crime. Balaam is both a Prophet and a soothsayer; a professional soothsayer it would seem, of whom Yahweh makes use to proclaim even in Moab the glorious destiny of the Chosen People, when He was about to lead them into the Promised Land (Numbers 22-24).
But the two greatest figures of prophecy between Samuel and Isaiah are Elias and Eliseus. Yahweism was again endangered, especially by the Tyrian Jezabel, wife of Achab, who had introduced into Samaria the worship of her Phœnician gods, and Israel's faith was tottering, as it divided its worship between Baal and Yahweh. In Juda the danger was not less menacing, King Joram had married Athalia, a worthy daughter of Jezabel. At that moment Elias appeared like a mysterious giant, and by his preaching and his miracles led Israel back to the true God and suppressed, or at least moderated, their leaning towards the gods of Chanaan. At Carmel he won a magnificent and terrible victory over the Prophets of Baal; then he proceeded to Horeb to renew within him the spirit of the Covenant and to be present at a marvellous theophany; thence he returned to Samaria to proclaim to Achab the voice of justice calling out for vengeance for the murder of Naboth. When he disappeared in the fiery chariot, he left to his disciple Eliseus, with his mantle, a double share of his spirit. Eliseus continued the master's work against the Chanaanite idolatry with great success, and became such a bulwark to the Kingdom of the North, that King Joas wept for his death and took his farewell with these words: &quot;My father! my father! chariot of Israel and its horsemen&quot;! Not all the Prophets left their oracles in writing. Several of them, however, have written the history of their times. Gad and Nathan, for instance, the history of David; and Nathan that of Solomon; also Semeias and Addo the annals of Roboam; Jehu, son of Hanani those of Josaphat.....Is it possible that the historical books of Josue, Judges, Samuel, and Kings were called in the Jewish canon the &quot;earlier Prophets&quot; because of the belief that they were written by the Prophets or at least based on their writing? To this query there can be no solution.
(2) Prophetic Writers — The prophetic books were entitled in the same canon the &quot;later Prophets&quot;. Gradually the custom of calling their authors the prophetic writers crept in. There are four Greater Prophets, that is those whose works are of considerable length. Isaias, Jeremias, Ezechiel, and Daniel, and twelve Minor Prophets, whose works are briefer—Osee, Joel, Amos, Abdias, Jonas, Micheas, Nahum, Habacuc, Sophonias, Aggeus, Zacharias, and Malachias. The Book of Baruch, which is not included in the Hebrew canon, is united in our Bibles to the Book of Jeremias. The ministry of Amos, the most ancient perhaps of the prophetic writers, is placed about the years 760-50. Osee follows him immediately. Next comes Isaias (about 740-700), and his contemporary Micheas. Sophonias, Nahum, and Habacuc prophesied towards the last quarter of the seventh century. Jeremias about 626-586; Ezechiel between 592-70. The prophecy of Aggeus and in part that of Zacharias are dated exactly in 520 and 520-18. Malachias belongs to the middle of the fifth century. As for Daniel, Abdias, Joel, Baruch, as well as portions of Isaias, Jeremias, Zacharias, their dates being disputed, it is necessary to refer the reader to the special articles treating of them.
(3) The Prophetesses — The Old Testament gives the name nebî'ah, to three women gifted with prophetic charismata: Mary, the sister of Moses; Debbora; and Holda, a contemporary of Jeremias (2 Kings 22:14); also to the wife of Isaias meaning the spouse of a nabî'; finally to Noadia, a false Prophetess if the Hebrew text is accurate, for the Septuagint and Vulgate speak of a false Prophet (Nehemiah 6:14).
(4) Cessation of Israelitic Prophecy — The prophetic institution had ceased to exist in the time of the Machabees. Israel clearly recognized this, and was awaiting its reappearance. Its necessity had ceased. Religious revelation and the moral code expressed in Holy Writ were full and clear. The people were being instructed by the scribes and doctors—a living magistracy, fallible, it is true, and bound overmuch by letter of the law, but withal zealous and learned. There was a feeling that the promises were about to be fulfilled and the consequent apocalypse increased and intensified this feeling. It was not unfitting, therefore, for God to allow an interval to elapse between the prophets of the Old Covenant and Jesus Christ, who was to be the crown and consummation of their prophecies.

At the end of Public Revelation which ended with the death of the last Apostle prophecy will continue within the Church, but the Church does not accord the title of prophet to them.
Immediately after Paul’s oft-quoted 1 Corinthians 13 pericope on the importance of integrating love into every gift of the Spirit, he encourages his community, “Strive eagerly for the spiritual gifts, above all that you may prophesy” (14:1).
Prophecy is the normal biblical way of surfacing God’s will in one’s life. Though Catholics look to the hierarchy and Protestants open their Bibles to discover God’s plan for them, the people who gave us our Scriptures obviously had neither the Bible nor a hierarchical structure as we know it to fall back on.
They believed God had blessed certain individuals in their midst with a special gift of knowing God’s will in specific situations. People of faith were expected to find those insightful persons, listen to them and then carry out their message.
Paul was convinced that prophecy is essential for any Christian community. If you don’t have prophets, you’ll be walking in the dark, not knowing what direction to go.
The gift of prophecy is very much alive within the Catholic Church.

Father Michael Scanlan’s 1976 Prophecy and the Events of 2020
The Gift of Prophecy
Those who, like Father Scanlan, who died in 2017, speak such messages are believed to have the gift of prophecy, as described in the New Testament Book of Ephesians, Martin said. This gift and others are given for the building up of the Church, he continued, and, although associated with the Charismatic Renewal movement, they are supposed to be part of the normal life of Catholics.
Martin has given prophecies, as well, including one in 1975: “Days of darkness are coming on the world, days of tribulation. ... Buildings that are now standing will not be standing. Supports that are there for my people now will not be there.”
However, Martin said he considers Father Scanlan’s message to be more specific in its references to the crime and chaos that many Americans are seeing on their streets in the wake of the death of George Floyd and in the mention of churches here and in other countries being closed because of the coronavirus. Martin said the references in the prophecy to dependence on the institutions of schools and parishes also are significant, as dioceses across the country struggle financially.
But he said there are important lessons to be learned as Catholics and others experience the events cited in Father Scanlan’s prophecy. He said during the coronavirus shutdown, when his parish was having online Masses, his pastor pointed out the four ways Christ is present: in the Eucharist, in the person of the priest, in the word of God and in our souls.
When the first two aren’t available, Martin said, Catholics can develop an understanding of and appreciation for the other two. “Those are important things we could learn during this time, but a lot of us haven’t learned it yet. We’re so dependent on the structures of the church building that it will be hard if those are taken away again.”

More can be seen here:

Pope Benedict XVI Predictions for the Future of the Church (YouTube video)

As for the  teachings of the Church of Jesus Christ of Latter-day Saints, the term Prophet is quite different.
The General idea of Prophets in Biblical times is somewhat (not exactly) similar to the views of Catholics in this regards. However the similarities end there.

As members of The Church of Jesus Christ of Latter-day Saints, we are blessed to be led by living prophets—inspired men called to speak for the Lord, as did Moses, Isaiah, Peter, Paul, Nephi, Mormon, and other prophets of the scriptures. We sustain the President of the Church as prophet, seer, and revelator—the only person on the earth who receives revelation to guide the entire Church. We also sustain the counselors in the First Presidency and the members of the Quorum of the Twelve Apostles as prophets, seers, and revelators.
Like the prophets of old, prophets today testify of Jesus Christ and teach His gospel. They make known God’s will and true character. They speak boldly and clearly, denouncing sin and warning of its consequences. At times, they may be inspired to prophesy of future events for our benefit.
We can always trust the living prophets. Their teachings reflect the will of the Lord, who declared: “What I the Lord have spoken, I have spoken, and I excuse not myself; and though the heavens and the earth pass away, my word shall not pass away, but shall all be fulfilled, whether by mine own voice or by the voice of my servants, it is the same” (Doctrine and Covenants 1:38).
Our greatest safety lies in strictly following the word of the Lord given through His prophets, particularly the current President of the Church. The Lord warns that those who ignore the words of the living prophets will fall (see Doctrine and Covenants 1:14–16). He promises great blessings to those who follow the President of the Church:
“Thou shalt give heed unto all his words and commandments which he shall give unto you as he receiveth them, walking in all holiness before me;
“For his word ye shall receive, as if from mine own mouth, in all patience and faith.
“For by doing these things the gates of hell shall not prevail against you; yea, and the Lord God will disperse the powers of darkness from before you, and cause the heavens to shake for your good, and his name’s glory” (Doctrine and Covenants 21:4–6). - Prophets

Every President from Joseph Smith to Russel M. Nelson is considered a Prophet in the viewpoint of the Church of Jesus Christ of Latter-day Saints (LDS).