After an accident, you may not have the other person’s name. Sometimes the other insurance company will contact you and give you the name of their “insured” but this may not be the name of the driver of the car. How do you get it?
First, you can get it from DMV through the SR-1 route. (Here is my prior post on this.) You file an SR-1 if there is more than $750 in property damage or if someone is injured. Then, 20 days later, send in an SR-20 with a check for $20 (although check the rate). It is not the most efficient way of doing it.
The second option is to send a letter to the insurance company and demand that they request the information from their insured. There is a case on point, Boicourt v. AmEx that says the insurer then has to ask. However, if they do or not varies.
The third option is to file a lawsuit and if you file in Superior Court, you (or preferably your attorney) will send out form interrogatories. One of them asks for insurance information.
Finally, attorneys can usually get this information and persuade the insurance company to release it.
The privacy laws for insurance are complicated but they are designed to protect you. But, when you are trying to get information, these same laws can frustrate you.