If I'm paying for damage I caused to another car out of pocket, does the other driver have to provide 3 repair estimates?

I rear-ended another car but the damage caused to both vehicles was very minor so we both decided not to file a claim through insurance. I am planning to pay for the damage to the other car out of my own pocket but the dealership where she took her car wants much more to repair the car than I think is fair. Is the other driver required to provide 3 repair estimates?

There is no requirement for a specific number of repair estimates. Unfortunately since you decided not to file the damage claim with your insurance company then it is up to you and the other driver to negotiate a fair repair cost. You could always refuse to pay the amount, but then you could face legal trouble. If you are concerned that the damage will be more than you can afford then you should seriously consider pursuing the claim with your insurance provider. Better safe than sorry. Best of luck!