First thing that must be recognized is that any endeavor done by the private sector rather than a public run enterprise is done better by the private sector. Note that in all cases, the publicly administrated organization has special benefits or an outright monopoly (think of the USPS which is protected by law to be the only entity allowed to provide mail delivery. Note that UPS and FedEx does not classify as such - just check the federal law in question).
Libraries could be all done and managed by private corporations that charges for membership and likely more than one such organization is in any city. For sure, the competition among them will make them work hard at adding services that will differentiate them from the rest of the competition. How about the poor? The wealthy usually buy the books brand new, they don't bother about going to the libraries, hence the majority of those who will be served are from the poor and middle class. If those companies wishes to attract them, they will need to offer a price low enough for the membership to account for their ability. The establishment where the libraries are could be owned by the city but managed by private companies, this is another alternative that is somewhat midpoint.
Central Park in NY. image from City of NYCity parks could still be owned by the city but managed entirely by a private management firm. Such firm would not receive a dime from the city and would not be taxed either on any profit in exchanged for maintaining the park with its primary purpose to serve the public a place to walk, jog, pic nic, with no fee to enter the park. How would they manage to get the money they need to maintain the park you ask? Well, some special food booth or other food court could be allowed by this management firm, again free of taxes, but for which these small restaurants would need to pay a monthly fee to the management firm to be able to operate in the park. You would be surprised how much better such park would be administrated.
In fact, a few city parks in the world are administrated privately and works much better this way.
To read more on this: