How to use CKEditor on the front-end?
On Input filter option (admin/settings/filters) you can check the user permission for each type of input filter you have on the site.

Then if you are using the WYSIWYG module, at admin/settings/wysiwyg you can check which type of editor is using each input filter.

If the WYSIWYG editor is working for anonymous user on one theme but not on the other, you could also check (with Firebug or Chrome DevTools) if there are Javascript errors. If that is the case, please, update this post with the error message.