I don't know how you properly define private ownership outside a legal framework of rules
Capitalism is an economic system which does not definitionally require government involvement. State-to-state transactions are examples of capitalistic trade where there is no over riding legal authority. It is true that undesirable effects such as monopoly, repression, and theft may occur without regulatory oversight, but that does not negate the capitalistic framework. I agree that state power may make capitalistic transactions more safe and predictable, but it is not a requirement.
State-to-state transactions are examples of capitalistic trade The fact that the trade occurred "between states" says nothing about it being capitalistic or otherwise.