I think Waqfs are by themselves legal entities, so the ownership question regarding waqfs can be solved by saying the legal entity of the waqf owns the property. Perhaps in common language, the word waqf is used to designate both the legal entity and the property (we say for instance: this real estate is waqf, while we also say: the waqf owns this real estate). In my opinion, this comes from the fact that ottoman law - based on Islamic law - looked at real-estate from the "legal categories" view, as established in the Ottoman Land law, where waqf was a legal category (shar'i) of real-estate, but the coming of European legal influence, especially continental law, threw some confusion, by adding the notions of "real" rights (droits réels حقوق عينية) separable from the property itself.
So - in order from the private sector to own waqf lands - it has to obtain decrees (according to each religion/state rules) from the religious authorities to allow the "istibdal" (since technically sale is out of the question). But, more easily, there is a certain number of contracts that can give the private sector total control of the real-estate without buying it, such as al Ijaratayn, Ijara tawila, Kadak, etc. Where the private sector is given long-lasting rights over the real-estate, including inheritance.