If you put the claim through your own collision coverage then your own insurance company would be paying for your totaled car minus your deductible. If instead you put the claim through the at-fault party's insurance carrier then normally their property damage liability policy would settle with you for the actual cash value of your totaled vehicle.

If your insurance company handles the claim then they may chose to subrogate with at-fault party's insurer to recoup the monies they paid out for your claim. You may ask if they will try to get back your deductible for you.

In some states though the claim process is a bit different. For example in Michigan you would only be able to get $500 from the at fault party (through the mini-tort) and then put the claim through your own collision coverage for your vehicle's damages or declare it a total loss.