Whoever is initially dispatched to the call by 911 also have a say. E.g. if 911 dispatches police, but the police feel they want an ambulance on scene as well, they can request that. And vice-versa.

There is a difficult balance to strike, in that police presence can escalate tension and make a situation worse, but police are also best equipped to protect other responders if a situation does escalate. It can be a tough judgment call