The exact same thing happened to me, waiting at red lights. The guy wanted to do it private, not go through the insurance, even though I'd already got his insurance details. It was a total nightmare DON'T DO IT!!!
We agreed on a local garage to do the work, all went together, the bloke, his wife and I. The quote was £450, all was agreed that the work would be done and he would pay.
When they took the back of the car off, there was more work than the original quote needing doing. They did it, called the bloke to come and pay for it. He said he would only pay the agreed price. The garage wouldn't release the car until the work was done. So I was faced with making up the difference in cost or not getting my car back!
I called my insurance company for advice, and thankfully, because we happened to choose an approved garage (total coincidence) they took over the claim and charged his insurance company the full outstanding amount. He went mental, phoning me saying how dare I take it through the insurance, it was going to put his premiums up even more, he needed to be able to afford to run his car for work, I was putting him out of work etc. It was frightening.
It turns out, he is a crap driver and has had to make lots of claims because he keeps having bumps and scrapes. His insurance was getting more and more expensive so he didn't want any more claims going through.
Seriously, just put it through the insurance. If you have his details, just do it.