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.
Are all the layers in your map public as well?
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