Chris explains why it's important to get the names and contact information of witnesses after an accident.
Should I get the names of the witnesses? Yes, you should!
If you get in an accident and someone comes up and is helping you, if you have the presence of mind to do so, absolutely write down the person's name. Get it on your phone. Anything you can do.
Witnesses are critical. It doesn't really matter that you know what happened because the other side might have their arguments that they know what happened. It may be a “he said she said" situation… but, boy, if you can get one unattached witness to tell the insurance company what happened - that does the trick.
So, yes, if you can get the name of that witness, find out who they are and, if they've left the scene and you have some way to track them down, do it. You'll be glad you did.