Depending on the construction materials (metal studs vs wood) sometimes not a lot. The ideal solution is to use 6" plates and headers and stagger the studs so that no adjoining walls share sheet-rock/2x4 connections, removing the physical coupling as much as possible. Thread some glass-wool insulation back and forth between the studs before installing dry-wall and you have a very cheap and effective sound barrier.
Edit: I learned this trick over 30 years ago from a builder who also owned a very popular low-cost motel with a dance-hall/music venue attached. Partiers can be petty noisy, and his place was really popular.