Insurance company guidelines differ but in general insurance carriers will require you to list all licensed drivers in your household on your insurance policy. If you have a 16 year old that obtained their driver's license you should inform your insurance carrier and your insurance company will tell you if he or she will need to be added. If your child is just starting the licensing process you may find that your 16 year old needs to be added at the permit stage or they may allow you to wait until they get their junior driver's license or whatever the next step is in your state's licensing (GDL) process.
An insurer is permitted to consider all resident operators of an insured vehicle in rating of an auto insurance policy. This includes your child, even if he or she has only a learner's permit or is newly licensed.
If you do not inform your insurance company about your child getting his or her license and they are not added at the correct time, according to your insurer's guidelines and the terms of your policy, than they may not be covered if in an accident. Or we have heard of situations in which the teen would be covered for the accident but only after you have paid the premiums that should have been paid for the child to be on the policy and properly covered.
If the insurance does not extend and the parents did not place the 16 year old on the policy as a driver but he or she was in an accident than typically both the child and parent would be liable for the damages the child caused.
Parents should not assume there is automatic coverage for a newly licensed 16 year old teenager driver. While their insurer may indeed automatically cover the child for a certain period of time until you add him or her to the policy as a driver, insurance company guidelines vary and so you should find out about your provider's rules and guidelines as your child starts the licensing process.
You can always compare prices here when adding on a 16 year old.