In Luke 22:19-20 it would appear that Christ simply gave them the bread (not fed it to them) and also it appears they passed the cup around. I have seen this done may ways over the years, sometimes we literally take a loaf of bread and break small pieces of it off (depending how many people are in attendance) more often, it is a wafer or a small piece of a cracker. I have never in over 35 years (at over a dozen churches) of taking communion had someone feed it to me. Maybe this is a protestant thing?
In 1 Cor 11:23-31 it doesn't really say how the bread was distributed out among the people. But again, these are simply symbols of Christs body and blood. I see no need to have a special way to receive the bread. (Other than in Holy reverence and self-examination).
I have never seen a reason for this written down anywhere, but in practice, we almost almost always pass the elements out in advance, then pray, then partake together. I suppose that is one big advantage everyone holding their own elements of communion. I myself wouldn't prefer someone else's finger handling my elements in advance. Especially fingers that have been putting these into a great number of mouths. It somehow seems rather un-sanitary to me. But that's just my preference.
It is refreshing to see the term "Eucharist" again. Most churches I have attended over the years simply call it communion. I suppose to avoid the ambiguity and negative connotations of the Roman Catholic "mass". Which is similar in some ways, yet very different in others.