How should I deal with an at-fault driver's insurance company?

​First, if you can avoid it, you should not deal with the other driver's insurance.  If you have your own collision insurance then report it to your own insurer.  This will no doubt work out better for you, as it will be easier (since your insurance company is looking out for your interests whereas the other drivers is looking out for their insured), it will be faster, and likely you will recover more.  Of course you will need to pay your deductible; however, that will be reimbursed to you when your insurer recovers from the at fault party.