A few states still give vehicle title information to private parties, most often through that state's Department of Motor Vehicles (DMV). Otherwise, give the license number to your insurance company or police department. They can access the DMV database without restriction.
Some Identification Obstacles
Not all states provide the name of a vehicle's registered owner upon the request of a private party. Washington State, for instance, specifically refuses such information when the search is related to an accident, such as a hit-and-run.
There are several reasons for this refusal, one of the more compelling being that providing the information can help a stalker or an abusive former domestic partner locate a victim. Increasingly, states tend to provide this information only to police departments and insurance companies.
How to Get Title Information
State Departments of Motor Vehicles keep comprehensive databases of vehicle ownership. When a state does release that information to the general public, it's available through the state DMV or another agency responsible for vehicle licensing.
Getting the information is different in each state. but begins with an internet search for the information from your state DMV or related agency. In order to avoid being overwhelmed by private party ads masking as state agencies, be very specific with search terms. For example "owner of vehicle with license plate ____" results in dozens of private party offers, some potentially bogus. Instead, search for "State of (state name) Department of Motor Vehicles title search."
However, in many cases, as in Texas, your search may conclude in a notice that the Driver's Privacy Protection Act (DPPA) prevents you from seeing that information.
A Way Around the Problem
Fortunately, although DMVs may not release the information to you, they have well-established lines of communication with police departments and insurance companies and will release the driver's name and other information to them.
If you have the license plate number of a hit-and-run driver, for example, giving that information to your insurance company in your accident report allows it to identify both the vehicle owner and that owner's insurance company through the DMV. At that point, the process no longer requires your involvement.
You can also file an accident report with the governing agency, either local police, sheriff's department or highway patrol. They, too, can see whatever information the DMV may have and take appropriate action.