The webmaps that are displaying the login screen have layers that reference items that are not publicly shared. The webmap works in the ArcGIS.com Map Viewer because you are logged in and have access to the items. When embedded, your login information is not available. Thus, even thought he webmap is public, the items in it are not.
The solution is to share the items you added that reference your map services publicly. If you don't want to do this, you can reauthor the webmaps and add your map service layers directly to the webmap, not based on an item.
We need to make a change to ArcGIS.com to let people know when they share an item publicly, like a webmap, that the item may reference other items that are not shared publicly, which you may want to make public. This is something we want to address in a future release.
Thanks,
Mike
