What is the argument in support of personal interpretation of the Scriptures? The Catholic church teaches that only the Church (Magisterium) has the right to interpret Scriptures whereas Protestant churches typically believe that anyone has that right.

What is the argument that supports the idea that anyone can and should interpret Scripture for himself or herself?
The Bible itself -- if you'll pardon my interpretation -- seems to support such a practice. 

In the Old Testament, the law is read aloud to the people (Deut 31:10-13; Josh 8:34f; 2Kgs 23:2; Neh 8:1ff) and the king is required to copy out the law so that he may know it and do it (Deut 17:18ff). In both cases, it is read "raw" and the people and rulers are to understand it and do it. Likewise, the Psalms are meant to be sung by the people, and Proverbs is in part a training manual in wisdom for princes to be read and understood.

In the New Testament, Jesus preaches directly to the people, the Apostles and missionaries preach to the people, and Paul intends that his letters be read aloud to the people (Col 4:16 -- possibly with the courier providing some additional clarification). 

Now these are authoritative teachers teaching, so one could argue they are similar in function to the Magisterium, but the "noble Bereans" are also commended for independently searching the scriptures to confirm Paul's teaching (Acts 17:11); Priscilla and Aquila, who are certainly not in the Magisterium, tutor Apollos (Acts 18:24ff); and Luke-Acts was written for the private individual Theophilus (Luke 1:1ff; Acts 1:1).

None of this is to say there is not a binding authority given to the church; there certainly is -- even in Protestantism (how else could church discipline exist as one of the "true marks" of the church?). But individuals are still encouraged to read, hear, understand, and apply the words of the Bible themselves.

In Richard Pratt's formulation of Reformed hermeneutics, IIRC, there are three sources of interpretation one seeks to respect: historical (the church of the past, tradition), community (the church of the present, your pastor and elders), and individual ("private interpretation", conscience, internal illumination by the Holy Spirit). In "remnant" situations, the individual interpretation may be the most correct, and God expects his people to abide by his true word, regardless of what the corrupt authorities say (cf. the OT prophets, and Jesus and John the Baptist vs. the scribes and Pharisees in the NT). That being said, going against broad agreement in interpretation across history and in one's present community should be done only with great prayer, humility, self re-examination, and trepidation.

A footnote on 2 Pet. 1:20 ("no prophecy of Scripture comes from someone’s own interpretation"), a contested verse in this context: "Prophecy" here is effectively "God's communication." Reading the surrounding verses shows (I think!) that the passage is concerned with the origin of a prophecy, specifically that it does not come from a private or merely human source. It is not addressing interpretation or understanding of that divine communication, which is what we're talking about in this question.