A few local state DMVs offer a free license plate check, along with a license plate finder or some other alternative to confirm license plate availability. The DMV's tools are often used to find a license number for a custom plate. However, is someone is tryint to use DMV's service to learn the details of a fellow car owner, he may have to involve law enforcement to obtain the needed information.
Aside from the DMV, it is very hard to find another trustworthy free license plate number check service because the DMV -- the government body in charge of issuing license plates for all cars owned by people in the United States -- does not provide current registration records to any public or private database. The DMV is only compelled to release protected information to law enforcement and licensed private investigators who can verify their need for the information.
Individuals may come across websites that offer free instant license plate searches, but these are likely false advertisements or scams. Generally, users are informed that the desired information has been found and is accessible if you pay a subscription fee. The problem is, there's no guarantee that they are receiving current driver and vehicle information because these are protected by the DMV.