I really wish these would go away. I assume the Continuing Anglicans do not have them; I hope the ACNA gets rid of them. In the ancient churches of the East and in traditional Anglicanism, the distribution of the Eucharist is limited to priests, bishops and deacons (in fact, in traditional Anglicanism, can deacons assist in serving the Eucharist? I think so, but am not sure). If there must be another type of minister, let us have Subdeacons to read the Epistles in larger churches. That is a traditional ministry which does not interfere with anything. Also, the rank of Chorepiscopi, although not traditional in the West, could be useful; these are glorified priests as it were with limited episcopal powers, who specifically can ordain Readers and other persons in minor orders and also do other minor episcopal functions, but who cannot ordain or as a rule participate in the ordination of priests, bishops and deacons.
My jurisdiction does not have this aberration. There are licensed lay readers. There are chalicists (typically postulants who have received the tonsure and some degree of liturgical training in preparation for ordination to the diaconate). There is not everyone and their cousin behind the rail trying to distribute the consecrated elements.
It occurred to me that where you find lots of these 'lay eucharistic ministers' the rail has disappeared. I once attended the RC church in Mass. that the Kennedy's are said to attend when they bother to go. That parish deployed a whole squad of these Eucharistic ministers -mostly women. We went up to receive communion because, hey, if you're going to commune a bunch of Democrat politicians why not commune some Anglicans too. But the chalice ran dry just before my wife got to it. They did not permit her to go to another station where there was still wine, she only got the host that day. She was annoyed.
I dislike partaking of the Eucharist directly from a common chalice; I prefer intinction and the use of a spoon, or better yet, intinction and receiving on the tongue the intincted bread in the Syriac Orthodox manner (this is actually the most sanitary way to do it, because the priest kind of pops the holy Communion into your mouth without touching you, or it). That said my liturgical piety is such that I believe we can only be harmed by the Eucharist as described in 1 Corinthians 11:27-34; if it is a valid Eucharist received appropriately and reverently, it cannot harm us. However, if it is not a valid Eucharist, or if we are guilty of not discerning the Body and Blood of our Lord, if we partake under the conditions we are told not to in the BCP, well, in those cases, it can harm us.
One of the things I love about the Continuing Anglican churches is they don't have laypeople distributing Holy Communion. My biggest gripe with the new mass in the RC Church is all the "extraordinary ministers". Put the altar rails back and return it at least to priest and deacon only distributing Holy Communion. It worked before back when there were way more people going to mass on Sunday so there's no reason it can't work now.
We don’t have any Eucharistic ministers in the REC (a jurisdiction within ACNA). Most ACNA churches I’ve visited don’t have them either.
Tridentine diocesan masses at St. Mary Magdalene in Camarillo, California (in the beautiful chapel, the crypt of which houses the Camarillo family) are a full house, every Sunday. It’s a beautiful site. Also the Anglican Province of Christ the King in Chico fills up. Orthodox churches also tend to get full on weekends. The Coptic parish I attended for a while was invariably overcrowded. It took 45 minutes to serve communion, in part because the Coptic church uses a very slow process where if there is only one priest, the men first receive the Body, then the women, then the men receive the Blood, and then the women (if there are two priests, or one priest and a deacon, both kinds can be given with one trip to the altar, one by each celebrant), and in part because the congregation was massive. About 100 kids and 300 adults crammed into a tiny chapel about 2000 sq.feet.