Normally if you are putting in a claim with the at fault party's insurance company (which in this case happens to be the same company that your car insurance is with) you do not pay a deductible. They would pay a deductible to repair their car using their collision coverage but in most states your property damages would be paid for by the at-fault person's liability policy which does not have a deductible.
If for some reason you have decided to put the claim through your own collision coverage, then you could discuss with the insurance adjuster about how the deductible will be taken care of. You might have to pay for it and then see if you can recover it from the at-fault party or through their insurance policy.
Both drivers in an accident having the same insurance provider is a common occurrence so most insurance companies have an internal system of how to deal with subrogation within their own company. It might involve inter-company arbitration or just the claims adjusters processing each claim working together to resolve any issues.
For information on how your specific insurance company will deal with the situation, contact your insurance agent or adjuster in the claims department and ask for information.
How To Deal With Your Insurance Company Regarding Claims is an article in our Learning Center that you may find of help.