Are there any private room for nothing?

Yes, there are three private rooms, one of them is large but the 3 are comfortable

There are two study rooms first come first serve. All these rooms are unavailable during the time we have free tax help at the library. For more information, visit www.vpslibrary.org

I think they have private rooms and if they do it should be free simply show your library card. But because it's a neighborhood library, you might want to call ahead to check availability.