You absolutely do not have to give a recorded statement to the other guy's insurance company. In fact, we strongly recommend against giving a recorded statement to the other guy's insurance company.
As for your insurance company, most insurance policies require you to cooperate generally and most specifically require you to give a recorded statement if asked. So, if your company asks you for a recorded statement you will probably have to give them one or risk jeapordizing your coverage.
Make Sure You've Heard and Understood the Question
Once you've agreed to give a recorded statement, the conversation will likely take place over the phone although it can occur in person. The most important thing you can do is ensure you've both heard and understood the question you are being asked. Once the investigator has asked the question, you should pause. Think about the question. Are you sure you heard it all clearly? Are you sure you understood the whole question and all of the words used? If you haven't, ask that the question be repeated or rephrased.
Control Your Answer
Answer the question you're asked completely and clearly (speak audibly, no nodding or head shakes, no "uh-huhs or uh-uhs"), but don't volunteer information beyond the scope of the question. And don't think out loud. If you're asked what time you started the trip that ended in the accident, your answer should be something like, "About 9:30 a.m." It shouldn't sound like this, "Well, let's see. I left the house early that day because my daugher had a dentist appointment. We were running late and I had to get her breakfast after her appointment and before school. So, we drove through McDonald's. That was about 9 a.m. Then I dropped her at school. I remember thinking how late I was for work. I probably left her school around 9:30 a.m."
Keep it Factual and Accurate
Don't accuse and don't accept blame. Only provide information that you know personally. Don't guess, speculate or assume. If you don't know the answer, then your answer is "I don't know." This is particularly true of questions about time, distance and speed. If you know how many seconds there were between the time you first saw the other driver's car and the time it collided with you, you can tell the questioner. But don't say, "A couple of seconds" if what you really mean is, "I'm not sure, it wasn't very long."
And if you're not sure about something, make sure you tell the investigator that you're not certain
Consider Each Question and Answer as Standing Alone
Make each question and answer complete by themselves so that, if read out of context, your answer accurately conveys your position. Straighten out confusion and clarify multiple meanings.
Keep Your Demeanor Polite
Be polite but not chummy. The investigator is not your friend. Nor do you need to treat the investigator like an enemy. Your job is to provide factual information in response to the inquiries you are given, no more and no less. Any other response, whether it be an effort to reach out to the investigator as a friend or a defensive or angry response will only get in the way of doing your job properly.
Tell the Truth
Remember the first rule of all insurance company communications: always tell the truth, even when it's uncomfortable...especially when it's uncomfortable.