In my non-expert opinion, this is expert advice.
Does your police report state that you are at fault?
You do have a police report, right?:umm
In December my old truck was hit (total loss) by an another driver (fled the scene, suspended license, no insurance) He was arrested later that very day after hitting 2 parked cars while fleeing police. Anyway, the insurance company of the owner of the car (girlfriend of the driver), got my phone number from the police report. They started calling me. They had questions they wanted answered. They were very impolite. I suggested they speak with (1) their insured owner, (2) my insurance claims agent and (3) read the police report. I hung-up. I stopped answering their calls. My insurance company wrote a big fat check for the total loss, even though my police report said I was not at fault, and is dealing with the other insurance company to get compensated.(probably to this day)
Short version; You do not have to speak to the other drivers insurance company. That will most likely be very unpleasant.
Let your insurance company, the one you have been paying for 20 years, do all that for you. They are experts.
Position your insurance company' claims agent in between you and the other driver's insurance company. :thumbsup