The local rental company will need to provide an estimate of the repair cost. They may not actually get the damage repaired (we’ve twice gotten rentals in Italy that had existing damage, which the rental company acknowledged and put on the paperwork when we picked up the car, but apparently wasn’t in any hurry to get it fixed), but they still need to tell you the amount it would take to restore the vehicle. Make sure it’s legible.
I once drove into a curb in Italy, and blew out a tire. The rental company sent a tow truck with a replacement wheel and tire, which I paid on the spot, and the driver gave me a scribbled handwritten receipt (the descriptions turned out to be not actually legible), and when it was submitted for insurance reimbursement, it was determined to be insufficient for supporting the claim. It wasn’t a huge amount, and I didn’t pursue getting better documentation for successful reimbursement. A bigger amount might have warranted more time and energy to do so.
In Norway last fall, we put a few scratches in the paint on the front bumper. They were just scratches, no dents, but Hertz Norway fixed the damage within 2 weeks of us returning the car, to the tune of $800. They were really good about e-mailing us a detailed repair report, in both Norwegian Kroner and the US Dollars that they charged to our VISA card. We also needed the rental booking documentation. Our insurance (provided by the same VISA) promptly credited us for the full amount. They also wanted photos of the damage, as well as photos (if available) of the area on the car before the damage occurred. It was all done via e-mail. A huge corporation like Hertz was able to provide what our insurer wanted. A mom and pop company may not be prepared to automatically do the same, so make sure you establish with them that you can and will be able to reach them again and again if more insurance claim documentation will be needed for you to submit a complete claim. More information will be better than less information.
Taking “before” photos of the rental car when first picking it up, as well as “after” photos when turning it back in, has become our standard practice now, as protection and proof.