So a student hit you eh? So is the sudent's Mommy and Daddy (M&D) paying for the damage to your vehicle? (since students usually do not have a lot of money) Does M&D know just how much a new bumper, tailgate, and hitch cost? Then there is the labor for body and paint work. This repair could end up costing a LOT more than the M&D of the student think.
You do realize that YOU could be left holding the bag for the cost of the repairs if M&D suddenly "forget" their kid had an accident and refuse to pay, right???? How do you think the insurance companies will react if you get them involved once M&D have their memory lapse??? That will get UGLY for BOTH of you.
I am not a big fan of insurance companies but in some cases they may be the best way to go. BTW, you do realize that some insurance companies do DROP coverage for people who fail to report accidents? That could get UGLY for YOU if your insurance company drops YOU. If there is a police report on this accident (since the other car was totaled and/or had to be towed) there is a remote possibility that the insurance companies may find out anyway. What if the body shop screws up and sends paperwork for your repair to your insurance company even though they were not supposed to?
Yes, I understand about trying to cut the other driver a break.
In my eyes you have more potential downside risk in this situation if you do not involve the insurance companies. Only you can decide if it is worth taking the risk.
Just my .02