Should I provide statements to my insurance company and the insurance company of the driver who hit me?

Your insurance policy typically requires you to give a statement to your insurance company as part of the investigative process. So, in all likelihood, you are required to give a statement to your insurance company. However, as to the at-fault driver, you're not required to give a statement to them, a recorded statement or otherwise. If you do, then you might want to seek the advice of an attorney to assist you in that matter.