There are a few things that are done, depending on which character has more dialogue/action:
If Big Bird has more meat in the scene, then Caroll will perform him, while someone, usually Jim Martin, will perform Oscar; but if Oscar has more to do in the scene, then Caroll will perform him, while Matt Vogel performs Big Bird.
Sometimes, Caroll will pre-record the dialogue of the character he's not performing, or, he'll do both voices during the scene, and rely on the other performer to be able to lip sync to his delivery.