You are correct; a deductible is typically only paid when using your own physical damage coverages (comprehensive or collision) to repair damages to your vehicle. When it is your liability coverage that has a claim placed on it by the other party whose car you damaged there is not a deductible to be paid.

If you were using your collision coverage for your vehicle to be fixed in the situation you described then you would have to pay out the deductible amount that you choose when you started up your physical damage coverages on the vehicle.

It is a personal decision if you want the other party to go through your property damage liability coverage with your insurance company or if you want to pay "out of pocket" for their bumper damages. If you pay out of pocket it is customary for the other party to get 3 estimates. Hopefully they all are close in price so that you know the estimated repair costs are accurate.