My apologies, my developer had a response and I forgot to post it for you:
Subscribers aren't allowed to see listings on the backend because that's what the role means--they can't edit posts. Other roles do see listings on the backend but BD tries its best to show only listings belonging to them and restrict administrative actions like approving the listing, changing Paid status, etc.
We had some issues in prior versions but the current version handles this correctly now.
It is not possible to show listings like that to Subscribers because posts aren't shown to Subscriber roles on the backend--this is a WordPress limitation, not a BD one. There are modules to change the permissions each role has, but I don't think this is a very good idea.
I agree that we should be able to show a user that he has some listings pending approval. We have a shortcode (businessdirectory-managelistings) that can be used inside a page to show the logged-in user the listings he has in the directory.
In any case, it is simply not possible to show these listings on the backend for subscriber roles. The screenshots posted just show this: the backend doesn't display listings. We can think of options on the frontend, but I'm not sure what's the best way to do this.