How to get registration info to be editable by user

I have a custom field on my registration page (phone number) that shows on the users profile in an accordion tab that only admins can see.
This works great for me as I also have their email (input at registration) show there.

I do not want to have other users see the contact info, but I would like the user to be able to edit the field (so they can edit their own phone number and as well as their email).

I have CB setup so they can edit their profles (both CMS and CB), but the fields I want do not populate on the user edit screen.

How would I go about doing what I need. I'm sure it is just a simple tweak that I'm missing.
Please let me know what additional info you would need as I'm sure my explanation is as clear as mud.
Thanks.

I may have found the culprit. I don't think the viewing rights were correct (they were set to Public) setting the rights to my member level (Registered) seemed to have got me on the right path.

If I figure out more this weekend, I'll update my progress.

Easiest way to do that is CB Privacy or CB Conditional so you can condition the profile view to only be accessible to moderators and user self. Either plugin can do this, but does so in a different way.

Aside from that you'd need to jump through some hoops since you can't have separate profile edit and profile view tab access permissions. You'd need to have the tab hidden on profile, create another tab for moderators, add Custom HTML fields to substitute in the fields from that hidden tab, but the users self still won't be able to see this on profile and there isn't really a solid solution for that beyond public visible tab with IF substitutions. Again, easiest to use a plugin built for doing this.