Who did Jesus intend to take communion? I believe the tradition of communion stems from the last supper Jesus had with his disciples.  Jesus had the last supper with just his 12 disciples, not all of his followers.  Does this mean that only priests should take communion, or should we assume that all followers should take communion, and if so why should we assume this from what is written?
It is pretty clear that when Jesus called his disciples together on various occasions was to teach them things that they were to pass on to the entire church, this is one of them.  Just as various commands on how we aught to live was only said to them, and not other disciples, so this practice was initiated with only them.

The text does not specifically say ‘only you should do this in remembrance of me’ nor does it say ‘tell everyone else to do this in remembrance of me’. Therefore we must turn to other scriptures to confirm that natural suspicion that Jesus must have been changing the meaning of Passover for Christians, just as all Jews used to eat the meal in remembrance of their deliverance from Egypt.

Turning to the Corinthian’s where the subject had come up due to their abuse of the practice we see clearly that all believers who sincerely believe in Christ should partake of the Lord’s Supper.


  Because there is one bread, we who are many are one body, for we all partake of the one bread.
  
  (I Cor 10:17, ESV)


Notice above ‘we all’ partake, that is all believers.

However Paul makes it clear than this should only be done in a right frame of mind by true believers and is not to be made light of like it was just an opportunity to party and get drunk.


  23 For I received from the Lord what I also delivered to you, that the Lord Jesus on the night when he was betrayed took bread, 24 and when he had given thanks, he broke it, and said, “This is my body which is for you. Do this in remembrance of me.” 25 In the same way also he took the cup, after supper, saying, “This cup is the new covenant in my blood. Do this, as often as you drink it, in remembrance of me.” 26 For as often as you eat this bread and drink the cup, you proclaim the Lord’s death until he comes. 
  27 Whoever, therefore, eats the bread or drinks the cup of the Lord in an unworthy manner will be guilty concerning the body and blood of the Lord. 28 Let a person examine himself, then, and so eat of the bread and drink of the cup. 
  
  (I Cor 11:23-27, ESV)


Notice above Paul is speaking to all the believers and saying they should all partake but not to do it in an unworthy manner.