Varies by state. Some (like California, I think???) are no-fault which means both parties pay (or their insurance companies do).
In any case, what you should have done is exchanged information with the other driver at the time of the accident:
– name & address
– driver’s license number
– license plate
– make, model, year, color of the car
– where & when
– take pictures if you can
At this point, what you want to do is call your insurance company. And notify them. You pay them money just for this kind of thing. Let them run the interference. Let them pay the bills.
BTW, in this kind of situation, if one person gives money to the other directly, then the person who received money is likely no longer able to collect insurance money at all.
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The answer to your question is: No, that’s not true. Liability is not based on a police report. The insurance companies can decide liability based on the facts of the case.