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Jenn:
When you sign into AGOL and author a map, that is "your" map. You save it and then share it to "everyone" if you want the public to see it. When the public finds your map, they can view it and add more data layers to it, but they cannot save over your version of the map. They do, however, have the option to save a "copy" of your map as their own map. Once again, they cannot overwrite your map.
While signed into your account, share the map to everyone and then also copy the map's URL. Sign out of your account and paste the URL into a new browser tab. Your public facing map should open and you should see a "Sign in" button in the upper right-hand corner proving you are view the map not as yourself.
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