Any agreement between a private party and the body shop is between them.
Normally a body shop that works with an insurance company will not pay off or discount your deductible amount. This would seem unethical in that it would appear they would just over-inflate the estimate to cover your deductible amount so that the insurance company ends up paying the entire amount for repairs, which is not right since you agreed to a deductible amount in your policy.
If the insurance company did not over-inflate the estimate to cover the deductible (which is insurance fraud) but instead took your deductible amount off their profit it is likely they would then find ways to cut corners to make up for this amount they did not receive. We would advise against repair or body shops that would cheat insurance companies since they are also likely to cheat you as well.
If you read through your insurance policy you will see that it says that you are responsible for paying the first portion of a repair bill in form of a deductible. If the cost of repairs comes in less than the estimate, you are still responsible for the amount of the deductible. Any savings legitimately belongs to the insurance carrier. Often the amount of the deductible can be the only profit a body shop makes on a repair so if they discounted customers this amount it would be hard for them to stay in business for long.
When you purchased your insurance policy, you signed a contract agreeing that you will pay the first amount of the claim up to your deductible. Repair shops should not try to hide or pay your deductible for you. This would be considered insurance fraud and thus illegal by both parties in most states. Penalties for insurance fraud are usually quite severe.
For example the New York Insurance Department notes that notes that if by paying the deductible a repair company charges less for the services than the charges specified on the bill that is provided to the insurer, such company and the insured may be committing insurance fraud pursuant to N.Y. Penal Law 176.05(1) and N.Y. Ins. Law 403.
Insurance company and body shop appraisers know the cost of repairs and are rarely tricked by inflated or inaccurate estimates. Be wary of the repair shop that offers to save you your deductible. They may be taking short cuts in your repair, which could depreciate the value of your car. There must be a reason they are willing to involve themselves in violating your insurance contract and possibly committing insurance fraud depending upon your state's laws.
To find out about a specific situation and/or your state's insurance laws regarding this topic contact your state's insurance regulator.