I have a WebApp Builder app that is shared Public. It used to be accessible and when a non ESRI user tries to access it, it asks them to login (they don't have an ESRI account).
A way to check with what Brian has mentioned is to add a private layer to your map and click save. A popup should display a list of layers that need their settings configured to public and you can see if that includes more than the one you added for the test and change accordingly.
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