I think you'll find a large number of older, and not so old military aircraft are in private hands.
I remember going to see a display by "The Confederate AirForce" in Tampa about 15 years ago. Fighters, bombers...
At a display in England about 10 years ago, 13 Spitfires flew, all but one of them, provately owned. On the same program, a privately owned MIG 21 fighter was displayed, and another was being offered for sale in the static park.