Each state and coverage is unique in regards to this question. This answer relates to physical damage coverage (coverage for your car).
If you are not at fault, you can file your claim with the at-fault party's insurance company and you won't pay your deductible. If you file with your insurance company, you must pay the deductible and your insurance company will work to recover damages (including the deductible) from the at-fault party's insurance company. This is called subrogation.
Most policies have a subrogation clause that enables your insurance company to obtain reimbursement from the at-fault party's insurance company for the cost of repairing your vehicle. This is the amount your insurance company actually paid out plus your deductible amount.
If your state has a Comparative Negligence Law then the fault for an accident is not always placed squarely on one participant. After determining the percentage of fault by each party, the compensation by the insurance company is adjusted accordingly. Allocation of fault is made by negotiation between the insurance company and the claimant. There are state-by-state variations in how comparative negligence works.
If the other company accepts 100% liability, then reimbursement is for the entire deductible amount. If the other driver involved in the accident is partially liable for the accident, their carrier will pay a percentage of the collision payment, and they may not reimburse or pay your deductible amount.