I don't see any way, but was just curious if it's possible. You know, to make it easier to find stories that we can add to right away without moderation. Also wondering if there's any way to easily differentiate public from private in story listings, like in the Recently Updated list?
I'm not aware of any way to search the Mode field. Hopefully if there is a way, someone will drop by and share it!
As far as I know, the About section of the story is the only place where the Story Mode is shown. (I've taken a brief look at the source code.) I'm also not really unhappy about that as public stories usually cause me more work than moderated ones.