Seriously, I've had 4 people hit me in the past 3 years. One totalled my car and I wasn't even in it, parked on the roadside.
2 hit me up the arse, one at a minimum of 50mph coming off the motorway and wrote it off, one skidded on ice around a corner onto the wrong side of the road and hit me oncoming towards him, around 20mph as we both saw it happening and braked, frontal damage but repaired.
And one was minor up the arse because he had a hyperactive child in the back.
Each time all had willingly given their insurance details, all apologetic, and for each one I went through MY own insurance. I gotta tell them anyway? They deal with it MUCH quicker. I pay for upgraded courtesy car cover so each time get a similar car to my own and drive it away.
Of course mine is unavoidable I drive my car for a living so can't afford to not claim for damage to be fixed. None of these could be avoided each time through my insurance, though each time sorted quickly, never waited longer than a month for the car to be fixed or the cheque to come through.....
My insurance care about 'me' they know they will get money back from the other insurance company. If I rang THEIR insurance companies I'd bet they would need pictures/dash cam/witnesses.
I needed a few pictures of my damaged rear from the guy who hit me coming off the M3 because he claimed I'd changed lanes and cut him off. Luckily I took pictures at the scene and took a witnesses number with a dash cam.
Your insurance is there for YOU. If someone hits you, their insurance is there for THEM. Which is when problems arise.
Go through your insurance only. I get I'm a bit of a rare case with many accidents in a short space (btw, never had whiplash 
that just adds to hassle (especially for fake claims!!)) but life goes on, after the last few years I'd never complain about a tiny dent, but also, drive a rather old car since the last accident!!