How do you find out who the insurer is when somebody hits you?

Get a copy of the police report. Usually your own insurance company can help you with obtaining this information. If you are unable to get obtain it, an attorney is usually the next step. In UK it is a legal requirement that you obtain the name and address of the other driver, and you have to give yours. Unless someone is injured or damage is done to a third party's property you need not involve the police. In practice it is a good idea to also give the name and address of your insurer, along with your policy number, to the other party. This speeds things up as you can give the information directly to your insurance company.