How do police/ambulance helicopters communicate with ATC?
The exact procedures will vary locally but typically ATC will give priority to police and ambulance helicopters that operate emergency services. 

If these helicopters are operating in controlled airspace, communication with ATC will be done by normal VHF radio. There is no need for a dedicated communications person, the pilot will be in contact with ATC.

Sudden manoeuvring of these flights is rarely a problem for ATC since these aircraft are operating under Visual Flight Rules. 

When chasing a car, the pilot remains responsible for keeping clear of obstacles and other traffic, therefore this requires a second person to keep a look out (either on the car or the surroundings)