Am I required to give the auto insurance company a recorded statement?
You are under absolutely no obligation to give the other driver’s insurance company a recorded statement. This is one of those lies that insurance companies tell. In most cases, the only purpose of a recorded statement in the couple of days after a crash is to try to get you on tape saying that you are not hurting that badly or that you think that your injuries are resolving. That way, they can use that recording to contradict what you and your doctors say about your injuries later in the case.
However, if it is your insurance company calling and trying to figure out how the accident happened, your insurance contract probably obligates you to cooperate with them in their investigation. If it is your insurance company calling, you should consult your attorney before making the decision about whether or not to give a statement.