My step-father was in the police for thirty years and my mother for eleven. Traffic officers are highly trained drivers and they have to pass very advanced tests. However, those officers driving ordinary patrol cars most definitely are not trained and they have the same driving qualifications and experience (often less) than anyone else.
My step-father was a terrible driver and he frightened me no end of times. Whether he was in a patrol car or not he would lurch in front of on-coming traffic, he never used his indicators and speed limits didn't exist! I remember him being disciplined after a member of the public reported him for driving a patrol car the wrong way down a one-way street. He just thought he could because he was on duty and in a patrol car!
Many police patrol cars are driven very badly indeed - no indicators, speeding etc. Unless they are going to an emergency they must obey the rules of the road like everyone else. But who's going to stop them and argue?