are there private "public library management" corporate or non-profit "chains"?

I can see here that state-run public libraries in America cost a lot of money and apparently many communities have trouble supporting them. Well, so could a private organization (whether for profit or non-profit) figure out a more efficient way to run a library (which might involve fewer employees with MS in Library Science and commensurate salaries) and then offer such communities library running services? Maybe there would be many different such library management organizations on the market, and if the local authorities get tired of one, they could kick it out and hire the competitor. The existing book collections and the buildings would still belong to the local state organization and be rented (or used for free under contract provisions) by the management companies.

Would that work? Would that be against the law? What would happen?