State laws differ regarding public and private information but if you need to know if a person has auto insurance you first should contact your local DMV to see if you as a private citizen can get this information. The DMV should then be able to tell you if you can get this information from them or another state agency.
If you have been hit by a vehicle and you have managed to get their license plate number and VIN number, perhaps this information was correct while their name and number was not, then you would also want to contact the police to make a report. It is no persons legal right to no if another person is insured unless there is cause for the information.
In some states if you have information on a car and need to know about the status of insurance on it to try and make a claim there is a financial responsibility unit or the police department. For example in Florida you contact the Bureau of Financial Responsibility if you have been in an accident and want to find out insurance information on the other party. In this situation you would need to send in the crash report and they should research and reply to you within 10 days.
In California you can request the insurance information by contacting the DMV's Financial Responsibility Unit and submitting a form.
In New York you fill out form FS-25 which is a Request for Insurance Information for NY Registrants Involved in an Accident and is available on the NYS DMV web site. You then send in the form with a $10 fee and the DMV will try to verify the insurance coverage of the vehicle.
If were not involved in an accident but believe that someone is driving without insurance or want to check and see if someone has insurance, your local Department of Motor Vehicles should be able to tell you if this is possible and give you the information or direct you to a state agency that can will give you the information you need regarding insurance on a particular motor vehicle.