The police did take statements and file a report.

With the first one, the person admitted rear-ending me and there was no drama.

With the second, the person who rear-ended me insisted that I had hit the car in front of me before they hit me. Nope. Luckily for me, the car in front of me only felt one hit - from me being pushed into them. But in that instance, no one was ticketed either. There was a little more drama as I waited to get a copy of the official statements and report to pass along to the insurance company.