A few weeks ago, I rear ended a car. At the time, the police officer on the scene said that there seemed to be damage of under $1000 and that myself and the other driver should try to settle between us. The car was a buick. I can't remember the exact model number. But as the officer had to really "search" for any damage done, I assumed that the repair would involve changing the bumper at the most.

I received a phone call from the other driver today and heard that the quote she got from her mechanic is $2600 and that there is a likelihood of discovering more damage underneath once they remove the bumper. I want to stress that I have no problem paying for the damage I caused, but it just seems like $2600+ for changing the rear bumper (for that's the only repair I can imagine necessary even though I am not an expert) on this Buick is too expensive.

What are my rights, if any, in this situation? Do I just have to pay whatever price they ask for?

Either that or report the collision to your insurance company. So you're either going to pay for it up front or possibly with an insurance rate hike. Regardless of whether you were charged as a result of the collision or not, the insurance company will find you 100% at fault.

Often the vehicle body will be out of alignment in a rear end collision. This isn't visible unless you specifically look for it - trunk lid is slightly askew, position of left rear tire is different from right rear tire, rear door no longer closes easily, etc. It costs a lot to fix this, and in fact insurance companies will total most vehicles with this type of damage.

I don't think there would be anything wrong with asking for additional estimates. Prices will vary from shop to shop.