Very few public libraries have qualified librarians working in them, certainly not doing day to day desk work. Those cuts were made years ago. Most public libraries are staffed by library assistants - and more power to them, but they're not librarians and they don't need a qualification.
You might have a youth librarian but they'll work across many libraries, or a team of stock selection librarians but they'll work in a central location.
If you see a job advertised as looking for a "librarian" then you'll most likely need a library qualification and you'll most likely not be working in a public library.
Our local authority public libraries are very stringent about library qualifications - you need a degree or postgraduate qualification, plus have, or be working towards, chartered status of the professional body CILIP. Very few actual librarians working for the LA though.