Speed limits are the jurisdiction of the entity that "owns" the road, so if it's a state highway, city street or county . . . that's the governing body. Some states have regulations that apply to all subordinant entities.

As to setting the limit, that also varies by jurisdiction. The typical situation is that there is a department of transportation, or roads or safety, that is charged with establishing or adopting standards. Public input, either via hearings or written comment, are usually required.