I would assume the laws are pretty much the same as those in the UK but you will have to make sure that they are though. When you're having driving lessons in your instructor's car, the vehicle is already insured for the learner and instructor and covers third party as well along with any damages to the car itself. An instructor will have comprehensive car insurance on the vehicle he's teaching you to drive in.
However many instructors will not want to claim through their insurance policy although it is already covered because they will lose their no claims discount and their premiums will go up at renewal. Therefore they will try to make the learner pay for any damages to their car. You should clearly refuse any attempt on his behalf to make you pay for the accident as you're paying for driving lessons which includes insurance cover. Irrespective of how many driving lessons you've had, if the instructor was willing to let you take his car for the driving test (or for lessons), it's his responsibility to get the car repaired.
And if he did not purchase adequate insurance, then that's his fault in the first place. A learner will make mistakes and may bump the car and that's a risk that instructors have to take and this is included in the prices for driving lessons.