Actually, when the kids understand what you’re doing, they usually can tune out. Of course, we do try to use our “quiet voices” and the readers, too. Also, if they know they have a turn coming up to read to you, that’s a plus for them to give others their time. If you move to where the child you’re reading with is sitting, you probably won’t disturb much at all. Or, you could have the reader come to you in a place “off to the side” of the action, if that would make you feel better.
Another idea you might try is before you release the kiddos to go to a round, tell them who you are planning to meet with during that time, so the kids have a heads-up.