That's not how it always works.
First, what the vehicle will bring at auction and what the insurance is willing to pay you for its salvage value are two different amounts.
Second, sometimes when you get rear-ended and pushed into another vehicle, your insurance company has to pay for the damage to the vehicle in front of you. It all depends on how close you were to the vehicle in front of you and how hard you were hit. This is why is is very important to stop in time to keep a safe distance between your car and the car in front of you. Of course, if a large truck plows into you, and you were a safe distance away, the truck driver is liable. On the other hand, if you don't leave enough room, your insurance will have to pay for the damage to the car in front. It may not sound fair, but I have seen it work that way.
Before you can say which insurance company is or should be paying, you need to know more details. For all we know, the person who caused the accident and the OP might be using the same insurance company. It might be that the other guy's insurance is paying for the damage to the rear, but the OP's insurance company is paying for the damage to the front, and the damage to the vehicle in front of the OP.