How would one go about writing an agreement for a payment of a car accident?

First try calling the police and getting a police report. They are the best weapon against anyone trying to change their story later. However, in the case where police won't show up or can't be reached for whatever reason, then take pictures of both cars involved if you can. Then get as much of the other drivers information as you possibly can, including insurance. Personally, private or written contracts I would not recommend because even if you meet the most seemingly trusted person around, there really is no guaranteed way they will pay for the damages to your car, or if even if they have thar kind of money for that matter. Sure you could write up an agreement and make it legally binding, and then sue them when they don't pay. But even a lawsuit will not guarantee you any money. However, to answer your question, consult a lawyer. Remember, whenever you enter into a verbal or written agreement with anyone, you are risking them changing their story later and it can complicate things so be careful.