Should I Contact My Insurance Company If the Other Company Is Paying for the Accident?

In most cases, I'd say the safe answer is "Yes, you should."  Since the other party's insurance company is paying for your losses (damages) in this scenario, it doesn't hurt to inform your company of the accident.  Check with your insurance broker, but reporting the accident should have no effect on you since the accident won't be considered "chargeable" to you.  That means the accident wasn't your fault, so it should not have a negative effect. 

Sometimes, the insurance of the other party is insufficient to cover your loss, so you might find yourself turning to your own company for extra help.  If you never informed them of the accident, and then you approach them weeks later, you might be out of luck.  Perhaps your contract with your company requires you to report accidents within a certain period of time.  You could find yourself in violation of your insurance policy.  In either event, it might be a good idea to talk to a personal injury lawyer to discuss what you should and should not be discussing with the insurance adjusters.  Oftentimes, what you say "can and will be used against you" in the future.   Always be truthful, but also be smart when it comes to these matters.  Also, I'd recommend discussing the matter with your insurance broker who should have your best interests in mind.