I have no idea about the standard of training and I am sure it varies by agency and state. The issue of pilots not having FAA licences is no different to the military pilots of most countries who go through their system and earn a military qualification. These qualifications are normally obtained through hard work, sacrifice and dedication just like the civil guys and the standard of training is generally higher and more varied since there is more money available, does not mean civil instruction is inferior. They fly in the same airspace and conform to the civil rules to do so, just because they do not have a CAA issued piece of paper does not make them any less qualified or less safe.
Personally, I find it very hard to believe that there is no formal training system for these agencies, simply because of US liability laws. Can you imagine the lawsuit if an agency helicopter crashes, for any reason, and injures or kills people on the ground? The pilot could sue the agency, even if it was blatant pilot error, and blame the accident on insufficient training.
Since the only pilots I know flying for a police agency are highly qualified with FAA licences, I am very curious to hear more about these rumours.