It is possible they both will need to be listed on the policy. It depends upon state laws, how the names are listed on the title and the guidelines of the car insurance company issuing the policy.
Some states require that the auto insurance policy is in the same name(s) as the registration. So, if the registration has both names on it, then both will need to be on the car insurance policy.
Some car insurance companies also require that all people on the registration be listed on the policy and rated, or excluded as drivers. Other insurance carriers will allow that only one person from the title and registration be listed on the car insurance policy.
How the names are listed on the title can make a difference to an insurance company on who is required to be listed on the policy.
With multiple owners, the title and registration will say "and" or "or." A title that states "and" may result in the car insurance company requiring that both parties be listed, where a title that states "or" may result in either party being able to be listed.
If you and another party are both on the title and registration of a vehicle, then first check with your state's Department of Motor Vehicles (DMV) to see if for the car insurance, you both need to be listed.
If the DMV says that either person can be listed on car insurance policy, and you only want one person on the policy, then you will need to shop around for a car insurance company that will allow this.