Usually, no. You are generally not required to give a recorded statement to the other driver's insurance company, and doing it too early can hurt your case. Talk to a lawyer first.
Here is the thing most people do not realize. The adjuster who calls a day or two after a crash sounds friendly, but that person works for the insurance company, not for you. A recorded statement locks in your words while you are still shaken up and before you know the full extent of your injuries. Neck and back injuries in particular can take days to show up, so a casual "I'm okay" on tape can come back to bite you later. See our pages on neck injuries and back injuries.
California is a pure comparative-fault state, which means the other side has every reason to get you saying something that shifts blame onto you. Even a small share of fault can reduce what you recover. A recorded statement is one of the main tools they use to do that.
Your own insurer is a bit different. Your policy may require your cooperation, but you can still have a lawyer help you handle it. Whether you are dealing with a car accident or another kind of personal injury claim, the safest move is to get advice before you hit record.
This is general information, not legal advice, and every case is different.
If an adjuster is pushing you for a statement, call Joe Crudo at (858) 622-7280 or reach out through our contact page before you say a word. The consultation is free.