While dealing with the insurance companies directly and trusting a qualified shop to do the work sounds like a wonderful plan, nothing is allowed to go quite that smoothly for me.
Both my own and the at-fault person's insurance company made the process as difficult and slow for me as possible, which led to me having to sue the driver directly. Neither insurer (State Farm and Geico) wanted to deal with the cost and complication of a 2015 VW being considerably damaged within 24 hours of being brand new - so they pushed me off in order to make it each other's problem.
Just to give you an example of how interested in making you whole insurance companies are... both my own and the at-fault insurance company had agreed to pay for about $6,500 in damage. I went to court and proved over $11,000 in damage which was granted to me in full.
To make matters worse the body shop that was storing the car during the time this was all being sorted out managed to do an additional few thousand in damage while it was in their care. The result of this is that no body shop will be trusted with any of my cars ever again, and I'm taking a few days off work to make use of the body work and paint equipment I own to fix it myself.