How this would be specifically handled would depend on the state, the carrier, and the events of the incident. Normally however 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. The at-fault driver would pay a deductible to repair their car using their own Collision coverage but in most states the damages to your vehicle would be paid for by the at-fault person's Property Damage Liability policy which does not have a deductible for you to pay out.
If for some reason you have decided to put the claim through your own Collision coverage, then you could discuss with your claims 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 that person's 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. Perhaps some insurance companies or in some areas of the US this is referred to as cross insurance as you did in your question however we are not familiar with this term being used to describe this type of claim (where both drivers are with the same carrier).
In your auto insurance policy declarations it should outline what to do in the event of a claim. The policy terms are per policy and the deductibles to be paid are outlined in the policy declarations. The insurance company will pay per the policy declarations, generally a deductible, if one applies is paid by the named insured when they place a claim through their own Collision or Comprehensive coverage. The deductibles in the policy are agreed upon by the insured and are only due for certain coverages.
These are the types of things your policy should tell you, again there should not be a deductible due if you are placing a claim through the at-fault party's Property Damage Liability coverage. The at-fault party however will owe their Collision deductible for the repair of their vehicle. There are not special contract exceptions when it is the same carrier for both parties in an accident regarding deductibles that are or are not due.
If you want to know more about your specific car insurance policy read through the terms of it and then contact your agent with any questions you may have. If you are not satisfied with your current insurer than shop around with us for affordable auto insurance.