I know of no general terms for songs needing more than one voice. There are many variants on the kinds of songs you describe and "challenge-response" is as good a general term as any. Other descriptions are "sing-along", "lead and follow", and there may be some arcane terms used by Lomax or other collectors for particular areas where this type of structure serves a practical purposes, as in long-haul chanteys, track-lining, rock-breaking chain gang work, and some kinds of lumber work. Dunno of any official terms for them though.