I don’t see a problem with accepting a students’ friend request. And you can do so as a professor, and still make your profile as personal as you want it for your friends. How? Facebook has amazing privacy settings. You can make a friend list (all students, for instance) and then restrict how much of your profile they can see, and how much of your activity they can see. Students can see my basic work, school and contact info, but cannot see my status updates, photos tagged of me, or what my friends write on my wall.