If the story can't be shared publicly, then readers will need an ArcGIS account so their identities can be validated. If they have accounts in your organization you can just share the story with that Org and they will be able to see it. Other members of that Org will also be able to view them.
You could restrict viewing inside your Org further by creating a group, sharing the story into that group and choosing who in the Org can view that group. You can also use a group to share your story with people who are signed to other Orgs.
The above sharing methods require Org subscription accounts.
Public accounts (ie. free non-commercial use) accounts can view publicly shared stories but can't participate in Orgs or groups. (If you are using a Public account, a very low budget and definitely not recommended way to share a story map you create with just a few people you trust would be to give them your account credentials so they can log in as you to view it, but of course they could intentionally or unintentionally delete or modify your work, or start making their own cool story maps with your account, or forward your credentials on to someone else who isn't trusted).