Should I give a recorded statement to the other driver’s insurance company?

In most cases you are not required to give a recorded statement to the at-fault driver’s insurer. It is usually wise to decline politely until you have spoken with a lawyer. Adjusters are trained to ask questions that can minimize your injuries or shift blame, and recorded answers can be used against you later. Your own insurance company is different. Your policy likely requires you to cooperate, which can include a statement or, for PIP, an examination under oath. Even then, it helps to have counsel prepare you first.