For Private Lessons, instructors need to be associated with the subjects they teach in order to appear on the correct lists in the public site and in Admin. You can set up this association in two places.
First, make sure all your subjects are entered.
Individually - On the Instructor Page
Go to Find > Staff and enter the name of the instructor. Click Edit
Scroll to the Subjects field in the middle of the page.
Bulk - on the Subjects page
Go to Configure > Subjects > Subject Association table
Select the teacher and the subject(s) they teach. Click Save.