Not sure if any place is doing it for police, but there absolutely do exist private fire services. You can choose to buy a subscription to their service, or not.

Pretty much anyone who hires private security (most large businesses, celebrities, etc) or a private investigator is effectively hiring private police.

They won't be able to fine, arrest, jail, or recommend charges to a prosecutor, but outside of that they can do pretty much anything police would (and many things police would not, depending on how busy the department is).

Many event venues will even require hiring an off duty police officer as security when serving alcohol, for example.