Here’s my experience-
In the past 10 years I was involved in two accidents where damages were well over $1,000. In each case I asked the other drivers insurance company to provide documentation that I shared in 0% of the responsibility (one driver ran a stop sign, the other rear ended me at a red light). Both insurance companies readily complied.
My rates did not go up because of either one and when I switched insurance companies and they did a search on my driving record it came up spotless.
I’m not sure where you’re from but in NY I’m fairly certain that legally you’re required to report any damage to your insurer.
(Someone knocked off our side view mirror and didn’t leave a note, there was a security camera pointed at the parking spot and the security from the building showed me the video of the car that did it. They said that they can’t give me the video without a court order but the insurance company will take care of that for me. The insurance company said they won’t help me since I have a $500 deductible but said I legally have to give them all the information.)