Steve Horwitz shares an interesting article about Central Park as an "Ostrom-type" privately managed public space. These sorts of public spaces are quite common. Two that are close to my heart are Mount Vernon and Monticello. Horwitz points out that the condition of Central Park improved tremendously since they started being managed privately. Mount Vernon presents an opposite example - it was very managed poorly under previous private owners until new private owners - the Mount Vernon Ladies Association - took it over (the federal and state legislature declined an offer to sell it in the 1850s).
I don't think it's easy to extract a simple narrative from these things. We know there are collective action problems associated with maintaining public goods in private hands. Mount Vernon seems to demonstrate this. We know there are public choice and collective action problems associated with a commons and with state administration. Central Park seems to demonstrate this. My concern with private management of these things is that not enough of it will be done. What is managed will be managed splendidly (visit Mount Vernon or Monticello if you have doubts), but a lot will be lost and left unmanaged, and you'll never hear people talk about those losses (the seen and the unseen, anyone?). My concern with public management is precisely what Steve points out - that it will be done poorly and risks not being responsive to what people actually want.
Some people seem to think there's a big bright line on these things - that the government is bad for these public spaces or that the market is bad. Taking either of these views seems to require that we ignore basic economics. I'm with Ostrom, personally - I think humans have a variety of ways to manage these sorts of resources. We have governments that sometimes work and sometimes don't. We have markets that sometimes work and sometimes don't. We have non-government, non-market institutions that sometimes work and sometimes don't. I'm personally interested in celebrating and supporting what works, not staking out one solution for pillorying.