It depends on the cooperative, but the most common one, is that the cooperative owns the building and only residents can own shares. Those shares are only for sale to future residents and can only be bought and sold from and to the cooperative. Hence no free trading. Generally for older ones they tend to be rather cheap.

So yes the building is somewhat owned by the residents, but in many cases it is deliberatly turned into an awfull asset, which is unprofitable for the owners.