First: Unless you signed a contract with an insurance company requiring you to take your car only to a specified shop, you can choose where to take it for repair. But the shop still needs to work with the insurer to come to an agreed-upon price. If not, and they still fix the car, you may be responsible for whatever the insurer doesn't pay.
For more details on "diminished value", non-OEM parts, getting a rental car, etc., please see our "How the auto insurance claims process works" page.
If you're in a wreck and it's more than a fender-bender, you might want to see our "What happens if my car gets totaled?" page. From it:
You have the right to payment of the actual cash value of your auto, and to expect a prompt and fair settlement. Don’t be surprised if your “value amount” and the insurer’s “value amount” do not match. Be ready to negotiate with the insurer when this happens.
The page has a lot more information on how insurers establish the cash value of the totaled vehicle, what happens if you can't find a comparable vehicle, and -- we get this question a lot -- what happens if you want to keep your damaged car.
If you have questions or problems with an insurer -- we're the state agency that regulates the insurance industry in Washington state -- give us a call at 1-800-562-6900 or e-mail us at AskMike@oic.wa.gov.