Is the compensation you are referring to for damage to your car? Medical bills? Or, pain and suffering?
You can demand compensation from whoever you want...no one is obligated to give you anything. You can not just demand money from the person and/or insurance and expect them to give it to you. It doesn't work that way. There is a system and a process to follow.
If you have collision coverage, that should pay for your car damage. And your auto insurance or health insurance should pay for your medical bills. After your insurance pays your bills, they will go after the at fault insurance co. to recover if the at fault insurance co accepts liability.
If you want pain and suffering compensation you need to prepare a demand letter explaining how much you want and why. The insurance will then respond to your demand, and possibly offer a settlement amount. If you are happy with it, you accept it. If not, hire an attorney and sue the at fault party. They will then submit it to their insurance to defend them.
What are your damages/injuries? You really need to provide more information for people to help you.
Good luck.