If my car was backed into on private property, do I have to report it to the cops?
Or can we exchange insurance information and handle it with out the cops?

This depends on your locality. In some places the cops will not come if an accident is on private property unless there was an injury.

Depending on your state, and at times even the city or the shift in a city, if they police will come out to private property and even do a report.
Most states do not allow tickets to be given on private property with a few exceptions.

Now if you do just exchange info, you of course run the risk of them giving you false ID, or fake insurance info. And of course with no real investigation or report, it will be your word against thiers, so if you car was backed into, tomorrow, they will call their insurance and claim you backed into them.