Yes, but all happiness buildings (circuses, colosseums, theaters, stadiums) in all cities (puppeted or not) are Local City Happiness. Local City Happiness will be included in your overall happiness totals up to local city population. So, for example, if a 3-pop puppet builds a colosseum and has no other local city happiness buildings, the full 2 happiness will accrue towards your total happiness. If that same 3-pop city then built a Theater, only 1 additional happiness would accrue. When its pop rises to 4, an additional happiness will accrue.