Ability to modify User information

Is there a way to extend access for unprivileged users to allow them to be able to access and modify
information on the ‘About Me’ screen?

Not tried it myself but as an RT admin user have you tried going to Admin->Global->Group Rights, selecting the “Unprivileged” group and then under the “Rights for Staff” tab ticking the “Modify one’s own RT account” (aka ModifySelf)? Does that give you what you want?