No priest carried out the MEAL in 1st Century Christianity, which is what the Eucharist was called!  There WERE NO PRIESTS!  There were presbyters and there were bishops.  But NO PRIESTS!  

At no time did Jesus ORDAIN anybody.  He did not lay his hands upon anybody, did not anoint anybody with oil.  All of this came AFTER Constantine became emperor and the persecutions of Christians came to an end.  THEN, the church's structure was set up to model the Jewish priesthood.  The bishops became the high priests, and the priests resembled those of the Tribe of Levi---the priestly caste of Israel.  

Christ is present where two or three are present.  Jesus stated this.  If the time comes when there are NO PRIESTS---the gathered community could, together, read the words of the Consecration, and Jesus would be present to them in gathering of the people, in the Word, and in the Eucharist.  This is how it was in the early Church.