Property rights certainly requires central calculation and coersion, too.
Look at all the activities undertaken in the name of copyright and patent enforcement. This is "property rights".
Look at cable companies suing cities to not give citizens fast broadband, while they maintain their cartel. This is central enforcement.
A company like Apple can be privately owned. The app store is privately owned. Facebook, Google, Amazon are all privately owned. Disneyworld is privately owned.
The only reason you don't see central enforcement on a huger scale in privately owned fiefdoms is because so far in the real world, people have collectively banded together to have democracies.
Property is a monopoly right to EXCLUDE others from use of a resource. It can range from excluding YOU from changing the color of your FB profile to exclding you from peeing in some woods.
The real relevant thing is how the thing is governed. Democratically (eg employee-run) or command and control (top-down)?
These corporations are technically owned by shareholders in a public stock market. Collective ownership is a major feature of socialism.
Alaska has one of the first basic income checks in the world as the citizens are considered all shareholders in the natural resources of the state.
This "private ownership vs The Government" is a red herring.
Governments exist in all organizations. Rights are those things the organizations recognize as worthy of defending with force, if the peasants are lucky, they codify a consistent policy and enforce them.