If you took the person's insurance details you are entitled to inform his company that he/she and you have had an accident. You are reporting the incident. You are just informing them that you may have a claim against them in the future.
I was rear ended by a guy who was going out of the country for a few days, we exchanged details, I reported (not claimed) the incident to my insurance company, I also reported the incident to his company. I didn't lose my own no claims bonus at next renewal because I had only reported and not claimed. I had no communication with the driver. I got my repair quotation, got a letter from my garage (main dealer) stating what they estimated was the loss in value of the car as a result of the crash, added in my doctor's bill (just one), the cost of the hire of a replacement car while mine was being repaired and I submitted the whole lot to his company with a covering letter stating that I would be happy if the final total was paid for and no more claims would be forthcoming. I followed it up with a phone call. A cheque for the full amount that I had detailed was issued, I got my car repaired by the dealer and banked the amount that I received because of the crash depreciation. I had reported the incident to the gardai. I even had his number plate. From the moment we exchanged details I had no more contact with the person involved. My letter just stated that I wanted to break even. I wasn't looking to make a profit. But I didn't want/deserve to be at a loss because of an incident that wasn't my fault. His insurer's settled within two weeks.
The initial reaction of the person is to say they'll settle outside of insurance but the thousands it costs to repair, pay replacement hire car etc. makes it a crazy route to take. That's what insurance is for.
Report to his (and your insurance - emphasise REPORTING not claiming). Tell his insurers that you will be claiming from them.