I've found it done both as a call and response, and as a song where everyone sings all lines.
One of the other complicating factors of the song is that the tune for the first three "questions" is different, but beginning with four, the tune repeats. However, most people I have heard, sing the rest of the song with two alternating phrases, one that moves, and one that stays on the tonic note. AND they alternate those, always starting the phrase that moves. This means one time the tune for -- say -- five bright shiners -- will be the single note, and the next time down it will be the one that moves.
Is that clear?
Blessings,
Barbara