Thanks to all who have replied.
I would like to emphasize that I am not asking under what circumstances it is appropriate for people to eat antidoron. Rather, I am asking if it is appropriate for a layman (man, woman, or child) to take the antidoron and start giving it to people. It occurs like this:
Bob, a simple layman, has just taken Communion. He then immediately gets 5 pieces of antidoron and walks up to Ian and gives Ian a piece. Then he walks up to a visiting couple, and gives them two pieces. Finally he walks up to Mike and Jennifer and gives them each a piece of antidoron.
While Bob is doing this, a 6-year-old child named Mary has also taken 3 pieces of antidoron, and she starts wandering about the church, wondering who she will give it to. She bumps into her friend, Hannah, so she gives Hannah a piece. Then she decides that it would be a good idea to give a piece to Hannah's father. Finally, she decides to give a piece to each of her friends Eli and Michael. Oops! She has only one piece left. So she breaks it in half and gives half to Eli and half to Michael.
While Bob and Mary are doing this, at least 6 or 8 other laypeople are also wandering the church, handing out antidoron in similar fashion. This is all happening while people are approaching the Chalice.
Is this a good situation, or a bad situation, or a gray area? Further, were Roman Christians and their children doing this each Sunday, or is this a modern development?