Why is the miracle of raising the dead often noted as being done privately? Elijah takes the widow's son upstairs and later returns the son to the widow downstairs so its assumed she wasn't with him during the miracle.

Peter closes the door of the room as well.

Elisha is also alone with the child.

I recognize there are some cases where the raising was public - particularly when Jesus was involved.  However, I still feel as if there must be a theme and meaning here I'm missing.
Answering from a Charismatic Arminian perspective of Scripture:

This is an aspect of God's Glory being veiled. We see this principle in operation throughout scripture beginning with the expulsion from Eden in Genesis (the Paradise of God being 'veiled' - barred - to mankind, cf. Genesis 3:23-24) through to the exclusion from the New Jerusalem of 


  ...dogs and sorcerers and sexually immoral and murderers and idolaters, and whoever loves and practices a lie. - Revelation 22:15 NKJV


This is one of the principles that lie behind:


  Do not give what is holy to the dogs; nor cast your pearls before swine, lest they trample them under their feet, and turn and tear you in pieces. - Matthew 7:6 NKJV


Why does the Lord veil His Glory in this way? There are two main reasons:

1. God is looking for men and women of faith - veiling His Glory gives scope for faith to be exercised by those who diligently seek Him and press in through and beyond 'the veil'.


  It is the glory of God to conceal a matter,
  But the glory of kings is to search out a matter. - Proverbs 25:2 NKJV
  
  The Lord is constantly watching everyone, and he gives strength to those who faithfully obey him... - 2 Chronicles 16:9a CEV
  
  Faith makes us sure of what we hope for and gives us proof of what we cannot see...without faith no one can please God. We must believe that God is real and that he rewards everyone who searches for him. - Hebrews 11:1,6 CEV


Contrast Moses and his attitude to draw close to God versus the Israelites he was leading and their attitude to shrink back from God's presence (cf. Exodus 20:18-21) - the latter had seen many amazing miracles, but rather than draw nearer to the Lord in faith, they kept focusing on problems, grumbling and complaining and hanging back from the God that they feared but did not love; while the former was called a friend of God and encountered the Glory of God to such an extant that his own face shone with the reflected Glory which had to be veiled because the Israelites couldn't bear to look at it either. Referring to this incident, Paul encourages the Corinthian believers that:


  But whenever anyone turns to the Lord, the veil is taken away. - 2 Corinthians 3:16 NIV


ie it is the inward circumstances of the heart (particularly whether it is believing or unbelieving) that determine what someone 'sees', not the outward circumstances of miraculous signs and wonders.

2. The Lord is merciful, and tends not to give us revelation beyond our capacity to steward appropriately (with consequent judgment for failing to do so).


  20 Then Jesus began to denounce the towns in which most of his miracles had been performed, because they did not repent. 21 “Woe to you, Chorazin! Woe to you, Bethsaida! For if the miracles that were performed in you had been performed in Tyre and Sidon, they would have repented long ago in sackcloth and ashes. 22 But I tell you, it will be more bearable for Tyre and Sidon on the day of judgment than for you. 23 And you, Capernaum, will you be lifted to the heavens? No, you will go down to Hades. For if the miracles that were performed in you had been performed in Sodom, it would have remained to this day. 24 But I tell you that it will be more bearable for Sodom on the day of judgment than for you.” - Matthew 11:20-24 NIV
  
  47 “The servant who knows the master’s will and does not get ready or does not do what the master wants will be beaten with many blows. 48 But the one who does not know and does things deserving punishment will be beaten with few blows. From everyone who has been given much, much will be demanded; and from the one who has been entrusted with much, much more will be asked.” - Luke 12:47-48 NIV
  
  The Lord is not slow in keeping his promise, as some understand slowness. Instead he is patient with you, not wanting anyone to perish, but everyone to come to repentance. - 2 Peter 3:9 NIV


Peter was a man that received the incredibly significant revelation that Jesus was the Christ (both internally from the Father through the Holy Spirit, together with external corroboration on the mount of Transfiguration), but he initially wasn't equipped to steward this revelation effectively - he needed to be sifted and effectively broken (somewhat akin to Jacob's identity changing encounter with the Lord at Peniel - cf. Genesis 32), before he was ready to respond appropriately to what had been revealed to him. 

Contrast Peter's faith journey with that of Ananias and Sapphira or Simon the Sorceror who you can see as people who were caught up in the excitement of God doing great things (through the believer's extravagant giving and the outpouring of the Spirit on the Samaritans), and wanting 'a piece of the action', but failing in their attempts and being judged, because their focus on the outward mechanics of what was happening around them was no remedy against the unbelief and consequent self-focus that was in their hearts.

Similarly - and I believe of most relevance to the cases you're referring to - the situation in Acts 19:13-17 with the sons of Sceva is another example of people who focused on the outward mechanics (using the name of Jesus) and neglected the interior necessity of faith (in the one whose name was being used) with drastic consequences.

Now given this background, imagine if you will, an average person's response to seeing a manifestation of God's Glory through someone being raised from the dead without previously cultivating a heart to diligently seek God as per Hebrews 11:6, they like pre-sifted Peter will want to camp in the Glory, grab a piece of the action like Ananias and Sapphira or Simon Magus, or focus on the mechanics like the sons of Sceva and try to reproduce the same results without undergoing the necessary refinement of faith. It will only bring them judgment not blessing.

This is why in today's age most of the more notable miracles such as the dead being raised, the blind seeing, the deaf hearing, the lame walking etc. are happening in the 'third world' amongst people of sincere and desperate faith rather than those from 'developed' nations with their modern medicine who are largely rich and in need of nothing (cf Revelation 3:17). In particular, those who have developed theologies to explain the lack of miracles they see are largely shielded by the providence of God from witnessing outbreaks of His Glory in this manner as they are ill-equipped to deal with them - they would be at risk of adding to their unbelief the faith-(what measure they have)-shipwrecking-sin of blaspheming the Holy Spirit.

edit: A further excellent reason is suggested through comment and I include it as well: Even though God does sometimes (nearly always? - cf. Amos 3:7) use human agents as part of the process of working His mighty works amongst us, it is right and proper for those human agents to downplay and even obscure their own part in the process in order that all honour and praise goes to the only One worthy of it. This particular aspect of veiling the Glory was not necessary of course in Jesus' case as he could say quite frankly:


  that all may honor the Son just as they honor the Father. Whoever does not honor the Son does not honor the Father, who sent him. - John 5:23 NIV