Instagram introduced a new feature in September 2018 that lets you share other users' Stories via Instagram Direct, the app's private messaging tool for solo and group messages. Problem is, it works both ways, so whenever you add to your ephemeral Story, everybody that sees it can share it directly with anyone else.
When somebody is watching your Story on Instagram, they'll see the Direct icon (a paper airplane), which they can tap on to share your Story with one person, multiple people, or a group. So a bunch of Instagram users that don't follow you could be seeing your Story, and there's no indication that anyone has shared your Story with those non-followers.
On the plus side, your story in Direct messages only lasts as long as your Story lasts, so once your Story disappears, nobody will be able to view it in Direct messages. Still, if you don't want them to see it at all, you can easily disable the feature in your Instagram settings on Android or iPhone.
Start by going to your Instagram profile page in the Android or iOS app. Tap on the three-dash menu on the top right, which will bring up several options in a side menu. You want to tap on "Settings" all the way at the bottom. In the following Settings (Android) or Options (iOS) page, if you're on Android, tap on "Privacy and Security." For iOS users, just look for the Privacy and Security section.
Whether you're using Android or iOS, under Privacy and Security", tap on the "Story Controls" option. There will be a toggle here for "Allow Sharing," so just toggle it off to disable sharing as messages.
Once sharing is disabled for Direct messages, the Direct button on the bottom right of each of your Stories disappears for viewers. So now you can relax knowing that only followers and those that happen upon your story (if it's public) will be able to see it. Below you can see the difference between an account that allows Story sharing through messages (left) and an account that does not (right).
Additionally, you can even take it a step further by turning off Allow Message Replies from the Story Controls settings. Doing so will prevent anyone from sending you messages on your Stories — if better privacy is what you're looking for.