We do a singaround at a local coffee shop. By default I'm sort of the "prompter" when I'm there. Ours is go around the circle style for those who want to sit in the circle - sing/play, pass, or request. Those outside are welcome to singalong, but aren't invited to sing or play unless they sit in the circle.
I'm always amazed at the people in the circle -- and most of the same people are there every week -- who seem oblivious to the fact that its their turn and will want to stop and talk unless I, or someone else, pointedly says "OK, Joe it's your turn!" It's like they're perennial virgins - "who me, oh my, I just don't know, that certainly came around quick, etc !!" Sometimes I just want to whack 'em up along side the head and say "Did you not hear the person to your right just do a song !!" But that doesn't lead to a fun time.
So, YES, someone needs to be constantly "nudging" the organizationally impaired people in a singaround situation - it's human nature.