Or rather, private libraries don't exist called that. Private video rental libraries do exist - Blockbuster, Hollywood, etc. They aren't called libraries, but they are.
All of this reflection brought on by Lew Rockwell: Sell the Public Libraries. He's right, as usual.
Private libraries are not subject to the crazy political controversies that constantly afflict public libraries. Should public-library computers be able to access porn and hate sites? Should they carry Mark Twain? Shouldn’t they have a section designed only for blacks? What about gays and lesbians, who pay taxes to support the libraries. Why shouldn’t their interests be observed as well? But that offends other people who similarly pay for libraries.