Should you give a recorded statment? Well that depends upon who is asking for one. If the at-fault insurance carrier is asking for a recorded statement you are not required to provide them one regardless of what they tell you.
On the other hand if your insurance company is asking for a recorded statment you may be required to provide one based upon your insurance contract or poliy language. Most policies have language in them that you agree to help your insurance carrier investigate the claim. Each policy is different and you should speak with a car accident attorney if you have specific questions regarding your particular claim.