You also should notify the other driver's insurer of the accident. The insurance companies involved will then begin an investigation of the accident in order to determine who's at fault. Ask the insurance company adjuster for a copy of the complete claim file. A good way to notify your insurer is by calling your agent, especially if the accident is minor. Exclusive agents sell policies for one insurer only, and are forbidden from interfering in the claims process.
Independent agents are a different story.
Insurers make mistakes from time to time, and it's possible that you could be surcharged for an accident even though it wasn't your fault. Insurers aren't keen on giving out copies of claim files. If the insurer doesn't grant you a copy, file a complaint with your state's department of insurance and contact a lawyer. The lawyer likely will have to subpoena the insurer to produce a copy of the claim file before the insurer gives it to you. You have every right to seek what you deem as appropriate medical care.