If you mean private as in state, it's illegal via the Contract clause in the Constitution. Having 13 states making up monetary policy became enough of a cluster

If you mean private as in state, it's illegal via the Contract clause in the Constitution. Having 13 states making up monetary policy became enough of a clusterfuck during the revolutionary war that Congress had to explicitly forbid the issuing of any type of currency, so chances are good it would result in an even bigger clusterfuck if it was tried today.