Yeah that is a old iTunes issue, but I was hoping Apple Music wouldn't mind.
Basically, iTunes is locked to one user per OSX user profile. So, if you are using one generic login then the first account that logs into iTunes is the only account that can use it unless you de-authorize and reauthorize the machine, which is basically what you are seeing. The easiest way around this is to give everyone using the shared machine their own OSX login, instead of using one generic login, then you each have your own instance of iTunes running. That would also keep others from messing up your Apple Music "For You" suggestions should you ever forget to log out and they start playing music you hate.
If that can't be done, you could try an old trick using multiple iTunes libraries, but I haven't the slightest idea it that will work with Apple Music or not.
How to use multiple Apple ID's with one computer and iTunes:
http://www.imore.com/how-use-multiple-apple-ids-one-computer-and-itunes