I'm a librarian and I work part time in a public library. The oversight comes directly from the public that we see every day. I clearly remember one borrower who came in screaming about how offensive a particular book was. The book was very critical of Hillary. The librarian who selected the book is a dyed-in-the-wool liberal who is active in the democratic party. (Aside: I see all kinds in this small library).
Some libraries are unionized, some are not. My library is not.