My friend's car has some scratches, and the driver has agreed to provide a quote for the repairs by tomorrow, warning that insurance claims could increase both of their premiums.
I'm not quite sure what that means, normally it would be up to the driver of the damaged car to get the quote and let the person at fault know.
You may get differing advice on here, as some people are sticklers for the rules, but to me claiming on the insurance really depends on a few factors:
how much damage has been done?
what sort of policy your friend has?
what their excess is?
How did the collision occur? Was he parked or moving?
Can it be proved who was at fault? Witnesses etc?
you may find if it is just scratches that the repair is less than your excess in which case there is no point going through the insurance, or if there is damage to the panels/bodywork then obviously it would be more expensive and you need to know what sort of cost you're dealing with quickly because if you are going insurance route you normally need to inform them within 24 hours of it happening.
There is a reason the guy left the scene though, he may not have a valid licence (although the paper copies are still used by some people, and could be completely legit), he may not be fit to drive or he may not be insured, either way he was hoping to get away with it so don't hold out much hope he is going to pay for any repairs, he could agree to it then refuse to pay up and disappear.
Ultimately it's a judgement call with the insurance, but unless you have protected no claims and it can be proven that the other driver was 100% at fault then be prepared to have you pants pulled down.