Hi David,
You can check the ACL setting for the respective profile type here(http://take.ms/YrnJF). If the edit/delete own comments ACL is on, then users can only edit/delete their own comments.

Thanks Raymond, I have the same setup as you but default users still can't edit or delete their own comments, see attached..

Ok, just checked. Users can edit but can't delete comments..

Hey David,
I created a regular user to test this on your site and I am able to edit/delete my own comments(https://www.screencast.com/t/gDvLg7e6G61h). Am I missing something here? Perhaps you can provide us the access of the user account experiencing this so I can take a closer look?

Also, do note that since you only allow users to edit/delete own comments, they would not be able to edit/delete other's comments.