We used to have multiple Facebook pages before I was hired as the social media specialist. Today we have one main page and have absorbed or "adopted" all of the child pages under one umbrella, save 4. Our Cancer Center, Trauma Unit, Foundation and our medically-integrated fitness center all have separate Facebook pages. I have admin rights on all of them but our Cancer and Trauma leaders do most of the posting on their pages. i handle Foundation posts but their team does as well. The fitness center is administered by our marketing team and the company that runs the center for us.
All of our other social media sites (twitter, instagram, LinkedIn) are corporate sites.
Most of the PR team and a few of the Marketing team members have admin or editing rights. In addition, some of our HR team has admin rights on LinkedIn for posting career opportunity stuff.
We are a small enough organization that we haven't had a whole lot of after hours things to deal with. When there is a community event, we modify our work week to allow for coverage and also I am OK with them sending me a photo or two to post. We haven't had a problem with coverage so far.