Do English speakers say something like "he pushed him back", even if they know who this "he" and who that "him" is, referring to two different people with the same pronoun? Or would they say something like "he pushed that guy" or "that guy pushed him"?
Yes. It's common when someone's telling a story out loud, for example. Context, cadence, and emphasis can all help clarify the meaning. But it's common enough in writing too, when the meaning is clear from context, as in this New York Times article:
Mr. Shultz said the President was ''astonished'' when he told him about the plan in December.