The BehaviorEditorPart should only be visible for the "owner" of a WebPart. If the WebPart is in the .aspx file, or was added to the shared page, then the BehaviorEditorPart should only be visible when the page is in shared mode. If the WebPart was added by an individual user in PerUser mode, then the BehaviorEditorPart should be visible for that user.
It is a bug in the alpha that an anonymous user can see the BehaviorEditorPart. However, in the beta an anonymous user will not even be able to enter Edit Mode on the page, so this will not be an issue.
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