According to the Wikipedia article about ventriloquism and according to an account by a former ventriloquist, there’s is absolutely no special “breathing” involved in this art! The main difficulty lies in avoiding the consonants B, F, M, P, Q, V and W by mostly substituting similar sounds.

Still, this does not quite explain how this ventriloquist today could both play the flute and let his puppet sing at the same time. But maybe it is possible to simply “talk” while having a flute in your mouth. I should try this.