I have shared an item with the Public, but the different download options are not available.
The Settings for the item does allow the data to be exported in different formats:
If the user does sign into their ArcGIS Online account the different download options become available. This doesn't seem like the way the download options on the item details page of a publicly shared item should work and seems like a bug.
Hi Mary,
What you are experiencing is intended behaviour.
The export functionality does not export to your computer but rather exports to a new item in your content. Depending on the export format users will see a download button available on the item page for the exported item. Because of this, a user needs to be logged in to an account in order for a new item to be created.
There is an ArcGIS Idea that you can upvote \ follow that asks for this behaviour to be included in future releases.
An alternative you can explore is to create an Open Data site in ArcGIS Online.
Cheers