True, though to be fair most people never get to use private libraries. Or they used a library at their University that was technically private, but that gave access to the public as well. Public libraries are ubiquitous and very normal, while private libraries are the exception.
In America, public schools all have private libraries, reserved for attending students. (Maybe some operate as public libraries, but I've never seen nor heard of it.)
Furthermore, public libraries are not necessarily free. In America they virtually are all; fees only for late returns. But this is not globally true; in some parts of the world, libraries open to the public charge a fee for checking out books, or even require a fee for entry.
It's a normal elision, yes, we all picture a public library when we say "library". But "free library" isn't redundant or weird, because "public" is a modifier of library, not a trait.
People tend to call their personal library a "book collection" or the like, but it's a library, in just the same way that a Little Free Library is.
So most people who read have at least a small private library, whether they think of it in those terms or not.
