How is the claim handled when the at fault driver and you have the same insurance company?

This is never a good situation - it does depend greatly on the insurance companies policies however the process is usually much more drawn out. In a regular claim your insurance company is fighting with theirs to get the money for your car. In this case they are fighting themselves so the 'sense of urgency' seems to evaporate. Make sure you really push them and keep them on their toes - just incase. This of course does not always apply but it is quite common in claims that have the same insurer.