John,
I've talked with a lot of students about this, I'll go over what we have covered in the past.
For a few students riding their own seems to allow them to spend more attention on their riding, and in the end it's best for them to do so. For some, they are so worried about their own bike, they never fully relax and enjoy the riding/school.
On the other hand, I've had many students that had never ridden a properly set up bike, and have been amazed at what they got out of one that had been set up correctly. Occasionally I'll ride student's bikes and sometimes I get off the thing amazed at how they could ride the thing at any kind of decent pace at all! In fact, it's pretty rare that I get on a student's bike, and have it work anywhere near as well as ours. From the training standpoint, this is an issue, as riders often have confidence problems, and don't know that it isn't them, it's their bike!
In addition to that, how often do you get to ride another person's bike on the track, that is in virtually perfect shape, and you don't have to worry about riding it hard--as hard as you want.
There are a few other minor reasons: one is mechanical. If ANYTHING goes wrong with your own bike (though not likely, it does happen), you could be out of luck. With a school bike, we have tons of spare parts (and spare bikes) so this is not an issue.
Another is the hassle of transporting your own, having enough fuel, or riding the thing home when you are really tired.
Finally, seeing an instructor on the exact same bike, you know it is down to rider, and not equipment, and that can be a factor in training too.
This got longer than I thought, hope it answered your questions! Don't hesitate to pm me if you have anything else you'd like to ask.
Best,
Cobie