Autobody, Bodyshop, Collision Repair Shop, however you want to call it, it may be an unfortunate need for you at some point in your vehicle ownership. Much like detailing, there is a necessary vetting process to insure you are finding a reputable at least and high-quality provider. Like detailing it is not as simple as people realize. Many experiences are not great, as demands for cost cutting often compromise the quality of workmanship for you.
First do not trust the insurance provider’s recommendation or allow them to bully you into choosing their recommendation. They are only thinking about their budget. You can choose your shop.
How does the shop look? It goes without saying, whether you are grocery shopping, car related, or any of the above, we will gravitate towards a cleaner location. Obviously, the staff is not painting cars in their lobby. But those details can be red flags in my mind. In their working area, a surgically clean environment is necessary to prevent the painting from being compromised. When in doubt, I would get as much of a tour of the area as reasonably allowed.
Certifications and using quality replacement parts matter. A manufacture certified shop is necessary due to the fact that OEM parts for your vehicle may not be accessible otherwise. This is in addition to the fact a shop may not be qualified to repair your specific vehicle make (Nissan, Porsche, Mercedes, etc). OEM parts are generally going to be a better option than aftermarket parts. Ask if they use OEM parts. My mechanic shop uses OEM parts. It is more expensive all around. But I trust the replacement part/repair significantly more. Is the shop I-Car certified?
Ask local car clubs. Same concept applies to detailing here. There are quite a few public forums which people are going to quick to recommend a place based upon cheapest price, that is guaranteed to disappoint quite a few people. Those cheap options ultimately become a waste of money and a headache.
My final point. When in doubt, a locally owned smaller auto body shop is going to be a better choice than a chain shop. Those local shops with a family vibe will be more likely to check all of the boxes above. They will more likely focus on quality over squeezing margins. That will include quality parts, not rushing, and paying for necessary training. Whether they are more expensive or not, the value will likely be higher.