Brainpolice:
I think you set up a bit of a false dichotomy. Cooperative or common property can be "private". To use purely individual ownership as the criteria for "private property" ignores the larger-scale manifestations of private property.
The fact that it is an association of private owners does not make the association itself private property. Democracy is made up of private owners, but the powers of democracies are not privately owned.
While the argument that one can buy out an association of private owners is valid, that doesn't mean the association will demonstrate the same efficiency at producing goods that a purely private owner would, and thus the fact that the ownership can be sold to a private owner simply results in a private owner buying out the association and substituting it for some other administration once the market process unfolds.
This is similar to a corporation whose shareholders fight and bicker with each other over control. The inefficiency of the corporation lowers the shares' price and makes a buyout easier, but this means that the corporate constitution is less efficient than that of the buyer.