To my knowledge no chartered city or municipality in the U.S. is currently policed by private law enforcement. Upscale residential developments, or gated-communities, often have private security. Some large corporations maintain security staff to protect personnel and resources, and to prevent industrial espionage. I think most Americans see law enforcement as a function of government. Many people feel privatized law enforcement would invite corruption (as if police corruption's never happened). Many feel a private company's own self-interest (being profitable) would threaten civil liberty and justice. Maybe.
There are several privatized prisons here in the U.S. I've got mixed feeling about that. Prison rarely reforms anyone from what I've read, and despite rhetoric about discouraging people from wanting to return, recidivism rates in the U.S. are high. There seems to be very little incentive to keep people out of jail.
Countries including the U.S. and Great Britain regularly use companies that specialize in military and law enforcement for many overseas operations. MPRI and Blackwater are a couple of names. MPRI was very active in the Balkan conflict during the '90s. The company provided logistics and peace-keeper training to the Serbs, but was accused of training soldiers for combat, interrogation techniques and more. More recently, Blackwater received notoriety when some of its people allegedly killed noncombatant Iraqis in retaliation for the murder and torture of four American contractors, as I recall.
Anyway, as far as I know ARK is a bit of a reality stretch. For now.