How do I obtain someone's car insurance information?

A car rear-ended me and we didn't see any damage, so we exchanged info rather than contact the police. Upon closer inspection, the bumper was pushed in and the estimate is over $900. I made a police report after the fact. I have the driver's tag number, license number, phone number and address of the person. He is not returning my calls, and I want to try to file a claim, but my insurance agent said I need to obtain his insurance info. Thanks. I'm in Florida.