What is an appropriate amount to charge trainers for use of a private gym?

This depends on several factors. But unless you want to constantly have trainers leave to set up their own gym, you should consider making the fee at least slightly more attractive than going it on their own. Many such private gyms will make the rate lower for trainers that have more hours in the gym. Which is a common business model, buying in bulk is less costly per unit. Again, I would suggest a small twist to this. New trainers are usually struggling to get established. I suggest cutting them a break as well to help them get rolling. A month or two to help them get rolling will go a long way to creating some loyalty.

As to a specific amount, you need to consider what the average training session is selling for in your area. Then decide how much of that rate is reasonable for you to charge for the value of your facility. It is not a cut and dry question.