Hi UniTexts,
Admins can always edit or delete listings, regardless of registration status.
If you allow for anonymous users to post, it’s hard for those accounts to get back to their listings to edit/delete them. The way we do that is to create fake accounts for them (WP requires this for post type assignment) but since the user doesn’t actually register, they don’t have that access after posting it.
You can always assign a listing to someone manually as the admin under Directory->Directory and then they can edit it freely.
But if you want reliable editing behavior, we recommend requiring registration so an account is associated to the listing that the user can get to.