State laws differ on how to locate a driver and their insurance company by just their license plate number.
Typically after an accident you should exchange information with the other party and so should have not only their license plate number but also their insurance company name and policy number so that you can call and make a claim if that person was at fault in the incident.
If you failed to exchange information or were hurt and taken away by ambulance to the hospital and were unable to exchange information with the other driver than their information should be on the police report. To obtain a police report you normally must contact the station in which the officer was out of and request a copy.
If the police were not called to the site it may be more difficult to determine the owner of a car by their license plate number. Information laws differ in the various states within the US however in some you can contact your local Department of Motor Vehicles and request information on a car owner by giving their license plate number. There may be certain procedures to follow besides giving the reason you want the information to obtain such information.
In some states there are specific agencies set up to help motorist that were damaged by a hit and run driver or need the insurance information of another driver they were in an automobile accident with. These agencies may be a financial responsibility unit, an agency within the DMV and should help those that have the plate number of a driver in an accident, especially that of a hit and run incident.
For example in Florida you contact the Bureau of Financial Responsibility with your crash report and they will 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 one must fill out form FS-25 which is a Request for Insurance Information for NY Registrants Involved in an Accident and is available on at the New York State Department of Motor Vehicles office or on the NYS DMV web site.
State laws, especially privacy laws, do differ so you may have to go through a strict process to show you want information on the driver for insurance purposes or to contact them about personally paying for the damages they caused and not for nefarious reasons.
As time goes by there may be easier ways in which to confirm insurance on a car and/or driver. For instance in Connecticut they recently went online with a way for drivers to verify that cars are properly registered. No personal information, such as names and addresses, are disclosed.
To check the registration one must know the vehicle type (class code), such as passenger, commercial, etc. and the license plate letters and/or numbers. After submitting the information, the CT site will give an expiration date for the plate or indicate a problem with the plate or information entered. If a person's insurance is cancelled and the CT DMV has been notified and the owner has not responded with new insurance information their registration should be suspended so when putting in the person's plates for registration verification it should indicate a problem.
While Connecticut and other states do not have a simple way to check the validity of insurance on a vehicle yet, it may be coming in the future. Until that is possible, contact your DMV and explain the reason you need to check on the insurance of a vehicle you only have the plate number for. They should either be able to help you or point you toward the state agency that can.