Obviously the same that allow the police to do the same in their duties.
It is the nature of authority that the one being sent is under the authority of those sending them.
So then it matters not if it is "people -> police" or "people -> government -> police" the people are of greater authority than the police.
Unless you wish to assert that the government exists separately and independently of the will of the people.
The people grant the police power to enforce the laws, solve crimes and apprehend those accused of criminal activity. If during an apprehension they are fired upon they are not restrained from using deadly force.