If you search this forum for "organization" you'll find a couple threads with tables of organization. And a funny thing is that they are all different. Whoever has the money makes the decisions, although at the professional level,
AEA,
LORT, and local union contracts will define who is "in charge".
My opinion is that the director should be in charge: then its a function of style, communication, and "chain-of-command" of the show.
Joe