Not entirely correct. The city can send you to a doctor for a full physical or mental exam as part of a fitness for duty evaluation. They can get your medical history as part of that from the insurance company. You can refuse, but they can suspend you without pay or fire you unless you successfully challenge the basis for their ordering the evaluation. It could be as simple as their suspecting you caught T.B. or Hepatitis from someone you arrested, or signs that you are a chronic alcoholic.