If two people are listed as the homeowners do both of them have to apply for homeowner insurance?

Yes, anyone with an interest in the property should be on the homeowner's insurance to protect their "share" of the property. If there are multiple owners, they can all share one policy, but it needs to name all of them, otherwise, when someone makes a claim, the insurance company may refuse to pay for damage to something in the home that belongs to the uninsured owner.