I personally run one Facebook page and assist with another. I know from managing these pages that may people look to them for information and to connect. People use these pages to rate the organizations, provide feedback, and to find out what is happening.
Some libraries with Facebook pages:
Some libraries with Twitter accounts:
Here are some interesting reads about social media in libraries:
Social Networking: This is part of a report done in 2012 by the American Library Association
Social Media: Libraries Are Posting, but Is Anyone Listening?: This is an article from the Library Journal
This is a slideshow from Slideshare:
Social Media & Public Libraries from akcrow
Does your local library use Facebook or Twitter? Do you "follow" or "like" them?