It depends upon the living situation and who has custody of the child and the guidelines of the parent's insurance companies whether the minor must be listed on each car insurance policy or with just one parent's policy.
If the teen lives with one parent and only visits the other occasionally than the custodial parent would of course need to inform their insurance company once the child started the licensing process and add the child to their car insurance policy once when the insurer required it, which may be when the minor has a permit or they may allow the parent to wait until the child has their driver's license. The parent that the child visits should also inform their insurer of the licensed child that lives in a different household to see if they are required to add the child as a driver, occasional driver or not at all.
Now when teenagers split time between divorced parents on a regular basis it is a common question of which parent should include the teen on his or her auto insurance policy. Many insurance providers suggest the parent, who has custody of the teen the most, should add the child to his or her policy. In some cases, insurance companies say it is whichever parent has custody of the teen when the child is attending school. It is a simple question to both insurance companies covering the divorced parents so we suggest the parents ask their carriers about their specific guidelines regarding this type of situation.
If the same company insures both parents, the teen will normally be covered by both parents' policies, regardless of whether the teen is listed as a driver on either policy. That is because some insurance policies define "an insured" as a person related to you by blood, marriage, or adoption who is a resident of your household. Since not all policies are the same though the divorced parents with a teenage driver will need to read through their auto insurance policies' language and contact your insurance companies for details.
It can get complicated when a child of divorced parents lives with one parent and the other parent gives the child a car to drive and many of the other situations that come up with divorced parents who have a teen driver so it is important to involve both parents' insurance companies to make sure the child has proper coverage in case they drive all the vehicles they have access to in both households. It is better to check and make sure that the child does or does not have to be on the policy than to assume that if the other parent has the child on the policy that it will cover him or her using your vehicles.