This is how disease is spread grooming kits are not person specific they are horse specific! I'd be very very unhappy if you turned up with brushes that could have been god only knows wear and used on horses that could have any variety of skin conditions or infectious diseases! In fact you would be very quickly asked to leave the property
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That is a very good point.
We do have a policy at our barn and signs posted in a couple locations that grooming kits AND muck boots are not to be used elsewhere and most people leave them on property in our tack rooms or kept in the car - which is where I keep mine to prevent things from going "missing".
I still think students, at least serious students, should have their own grooming kit as long as it is explained about the health issues associated with taking them off site and using outside of the barn. The concern about people using them on other horses outside of the barn, is totally valid, but easily maintained even with the little kids - the parents being made aware and a stern warning of what can happen if not followed. To imply students shouldn't have their own grooming kits is a bit much. :)
Another thing to consider is that it's not just
grooming tools that can spread disease. Footwear, clothing, etc. I bet there are people who would never ever use a brush on a random horse... but then go visit a friends barn elsewhere in the same muckboots that they wear at their home barn. At our barn, we don't allow people to use muckboots elsewhere and they MUST be stored on site. (Any visiting vets and farriers are to clean their boots upon arrival.)
There was a recent outbreak here and it got bad enough that all visitors were restricted from the barn, no one could visit any other barn / show and we were told to even avoid tack stores unless necessary. A bit much, but we haven't had anything at our barn in many years. Knock on wood!