If both drivers involved in a car accident have the same insurance company, then the company will assign two different adjusters to two separate claims that will attempt to come to a liability agreement among themselves, and one insurance policy will make payments to another.
In case both drivers involved in a car accident have the same insurance company, the case will be split into two different claims, and the company will handle them as if they were dealing with another insurance company.
However, the two adjusters handling the case are not going to disagree with each other in public, meaning they will come to a liability agreement among themselves. Payments will be made from the responsible party's policy into the other. Depending on the company, the situation should benefit the damaged party as the company should be interested in keeping him or her as a paying customer, so they would probably offer fair compensation.