Some years ago I dinged a lady's car. It was my fault. I told her I didn't want to run through insurance, but I'd take care of it. Good thing she trusted me. She didn't have a body shop in mind, so we agreed to meet at one I knew the next day. We got them to write an estimate, that came to $1200. I wrote her a check for $1400, and said get the car fixed wherever, or keep the money. Hand shake, end of story.
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Yeah, I'd never leave the scene of an accident without a police report (although there are some jurisdictions that don't dispatch police to minor accidents) and a copy of the other driver's insurance information. If a driver wants to pay for the damage out of pocket, no problem, I can hold off on filing a claim for a day or two, but I know that I have all the information needed to do so if the other driver doesn't immediately come through. And if they did not have cash in hand the next day, I'd be filing a claim right away. There's just too much risk and too little benefit as the non-fault party.
With my Tesla, I'd likely want to have Tesla's body shop do the repair (we have one in my greater metro area) while most people looking to pay out of pocket for the claim would want to take the vehicle to a lower cost shop.