We just set the privates up directly with the coach, and pay them directly the day of the private, after it is over. They are allowed to use the gym and the equipment for them, but all the money goes to the coach, so I have never tipped, or really even thought about it.
I do NOT tip. At our old gym each coach set their own rate and you paid them directly. I don't know if they even paid a share to the gym or not. At our new gym the gym sets the rate; but you pay the coach directly still. It is my understanding that the coaches pay the gym $10 per hour for using the facility for a private, which still leaves the coaches with $40 in their pocket for that hour. Not bad money.
For ski lessons I do tip our instructors; but that is because I pay the ski school and they pay the instructors a small portion of what I had to pay.
My mom almost always "tips" when me and my sister do privates. She doesn't call it that though--she just pays them more then what they charge. I asked her about it one time and she said that coaches overwork and are underpaid so she doesn't mind throwing an extra $10 in when they do even more for us. It is a small starter gym with very good coach-gymnast-parent relationships.
I have been tipped at the end of the year before by one girls dad. Her mom never tipped nor did I expect it but dad decided to. At that time it was nice because I only made my hourly rate for privates at that gym but my hrly was $20/hr+ so I wasn't hurting.
Mom and kid were easy to deal with and she was a joy to coach and proved I could take a once a week cheerleader from scratch to layouts.
Every once in awhile a parent will tip, and I have to say it made the younger coach's day. She also ran over time with her private, which is one reason she tipped. It's definitely not the norm though. Bday parties, like dunno said, you tip for though.