There is a guideline to set up the BuddyBlog Pro user settings that allow users to delete posts from the frontend.

Here is a screenshot showing Deletion panel options:-

1. Allow users to delete their posts?:- Admin can allow users to delete their posts from the frontend.
2. Enable deletion for:- Only selected user’s roles can delete posts.
3. On deletion action? After deletion post will move in the trash or delete permanently.

Profile Tab Post Misc settings with BuddyBlog Pro.