Accident handling without insurance. What’s the procedure?

Good morning together, I’m still a bit through the wind and therefore ask for your help. Before on the way to work someone is driving me into the side. The driver has taken the blame so that we want to do everything without police and possibly without insurance. We exchanged data. On my car the bumper is scratched and minimally sealed, to this the fender is scratched somewhat. Now I am a little uncertain how the procedure is there an accident as soon as possible to the Versiche I wanted to go to some car paint shop earlier today to get an estimate. I would inform the accident opponent of the costs. If he agrees, I will have the money transferred and the paint work carried out on it. After that I will send a letter that the work has been done and no further costs are to be expected. If not, I would have the damage to their insurance insurance as soon as possible. g report. Is that right? The car went funny from the feeling after the accident. But that could only have been imagination. Anyway, there is nothing to see on the tire. I think/hope a car paint shop would also detect mechanical errors?