In our parish, we have the custom of making small prosphori available at the candle stand. Before the Divine Liturgy, the faithful can send them into the altar with a list of names for prayers for the living and the departed.
When the priest receives the prosphora and the list, he prays for the people indicated and takes a small particle of bread for each name and places it on the diskos, in the center of which is the cube of bread which will become the Body of Christ during the Divine Liturgy. This is done at the Proskomedia, celebrated before the service, usually while the Hours are being read.
If one cannot offer prosphora during the Proskomedia, one can request the bread at vespers the evening before or at the end of the previous Sunday liturgy.
After the Liturgy, the prosphora is then given back to the person who requested the prayers, and if you look closely at it, you should find that small particles had indeed been taken out.
The origins of this practice go back to the days when the faithful would bring the bread and wine to be used in the Liturgy.
The prosphora you receive back, which of course has not been consecrated into the Body of Christ, is part of the “antidoron”, not unlike the bread the faithful receive when coming forth to kiss the cross at the end of the Liturgy.
“Antidoron” means “instead of the gift,” and in times past, only those who did not receive Communion would take it at the end of Liturgy. The prosphora you receive is also an “antidoron” and it is appropriate to share it with the person or persons in whose name it was offered, especially if they are home bound or too ill to come to the church for the Eucharist.