I mean sure you could call it something else, like a "Private Defence Agency" or whatever colourful euphemism you want but it's still basically a state. —Jon Hillstrom
A private defense agency (PDA), aka dispute resolution organization (DRO) works for its subscribers. They might patrol the neighborhood in cars like modern-day cops and private security guards, or heck, maybe just watch via satellite or drone camera and send in a chopper when they see a problem. Or they may only answer direct calls; it depends on how they set up their business, which (if they're smart) depends on customer demand.
Its employees have no special rights (nor do those of the state, in reality; see RightVsPower; but they are protected in the state's courts when they assert said "rights" to do harm, invade property, and extort people).
I.e., since its customers have no right to trespass and invade another person's property, or extort non-subscribers for money to pay it - neither do its employees! If they break down a door and shoot the dog - they are liable for both door and dog.
They only have rights delegated to them. E.g., part of the contract subscribers sign might include the subscriber delegating any right to retribution for harm to him to the agency, or allowing it to search his property with documented cause (possibly paying a fee if the alleged crime or, say, stolen item, is not found). Without such delegation, they don't have a right to enter anyone's property but their own. No "magic" badges or uniforms.
On the other hand, the state asserts and claims rights to initiate violence, and does it frequently. They extort and coerce peaceful people. They get involved in peaceful transactions where they aren't invited, either to steal a cut or prevent the transaction entirely. They hand out permission to pollute even when property owners don't want it and should be paid damages.
The fundamental differences, then, are that (1) a defense agency does not initiate violence, or, if it does (2) it is liable for it like anyone else and can't escape on some bullshit "sovereign immunity" nor does it get to be the judge of its own actions.