We just launched our story maps and as far as I can tell all of our story maps and GIS content are shared with the public. However, when viewed on our website a pop up appears prompting viewers to log in to ArcGIS. I'm not sure how to fix this.
I normally see that when something is secured, You could have subscriber content in one of the maps or other items. Or it could be the usage of the short url, Org.maps.arcgis.com, Vs arcgis.com. Its also possible to use your browsers developer tools (shift+ctrl+I in chrome) of a different web debugger (I use Fiddler) to inspect what request is secured. If you wanted to reply with a link to your app we can troubleshoot that here.
When clicking your "completed projects" link I can see the sign in displays. I'm also running the web logging tools at the same time, and I see a flood of web requests. I note a couple item Ids display in the log. I'm able to use Item Syntax "https://www.arcgis.com/home/item.html?id=" with the ID number, and found this following Subscriber Layer was called on.
When this item opens as a map, you will be prompted to sign in as this requires an ArcGIS online account to access.
A simple fix is removing the layer from the public maps, but here is blog that covers the use of layers like this in public facing content.
Use Living Atlas subscriber content in public maps and apps
I hope this helps.
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