What's the best way to inform a supervisor of collaboration on papers with a different professor?
In some cases, a supervisor will be happy to have a student work on independent projects as well.  In other cases, the supervisor will be happy for the project to happen but want to be involved themselves as well.  In yet other cases, the supervisor will feel the project is a distraction or problematic for some other reason and strongly advise the student not to take part in it.

Which case you are dealing with depends critically on the details of the people and projects involved, and the best thing to do is for the student and supervisor to talk.  Don't make it a big scary deal, just let the supervisor know about the opportunity, say you're interested, and ask what their opinion is.