aseh, if two aidim come and say that ish ploni ate a kazayis chazir in his bedroom, he is liable for makos. The only differences between public and private relate to kanoim pogim bo, and the status of a mechalel shabbos in public, which has the din of a goy.
Beis din has essentially unlimited authority, as they can punish in horaas shaah even when halacha doesnt require it, to teach a lesson.