From the 2002 General Instruction of the Roman Missal (GIRM) approved for the USA, which can be accessed from
romanrite.com/girm.html :
“287. If Communion from the chalice is carried out by intinction, each communicant, holding a communion-plate under the chin, approaches the priest, who holds a vessel with the sacred particles, a minister standing at his side and holding the chalice. The priest takes a host, dips it partly into the chalice and, showing it, says, Corpus et Sanguis Christi (The Body and Blood of Christ). The communicant responds, Amen, receives the Sacrament in the mouth from the priest, and then withdraws.”
From the 1975 GIRM:
"117. He then takes the paten or a ciborium and goes to the communicants. If communion is given only under the form of bread, he raises the eucharistic bread slightly and shows it to each one, saying: The body of Christ. The communicants reply: Amen and, holding the communion plate under their chin receive the sacrament. "
“2. Rite of Communion under Both Kinds by Intinction
246. If there is a deacon, another priest assisting, or an acolyte present:
a) The priest hands the minister the chalice with purificator and he himself takes the paten or ciborium with the hosts. The priest and the minister of the chalice station themselves conveniently for distributing communion.
b) The communicants approach, make the proper reverence, stand in front of the priest, and hold the communion plate below their chin. The celebrant dips a particle into the chalice and, showing it, says: The body and blood of Christ. The communicants respond: Amen, receive communion from the priest, and return to their place.
c) The communion of those who do not receive under both kinds and the rest of the rite take place as already described.
247. If there is no deacon, assisting priest, or acolyte present: a) After drinking the blood of the Lord, the priest takes the ciborium, or paten with the hosts, between the index and middle fingers of one hand and holds the chalice between the thumb and index finger of the same hand. Then he stations himself conveniently for communion.
b) The communicants approach, make the proper reverence, stand in front of the priest, and hold a plate beneath their chin. The priest takes a particle, dips it into the chalice, and, showing it, says: The body and blood of Christ. The communicants respond: Amen, receive communion from the priest, and return to their place.”
“4. Rite of Communion under Both Kinds Using a Spoon
251. If a deacon, another assisting priest, or an acolyte is present, he holds the chalice and, saying: The blood of Christ, ministers the blood of the Lord with a spoon to the individual communicants, who hold the plate beneath their chin. He is to take care that the spoon does not touch the lips or tongue of the communicants.”
All the official documents I am aware of direct that person receiving Communion to hold the communion-plate. None direct an altar server to hold it. So there is a problem if some are receiving Communion in the hand. What do they do with the plate?