I have a human face mesh that was created using the mirror editing, so one half of the face controls the other half as well, and only that controlling half is visible (highlighted for editing) in the edit mode. When I disable the x-axis mirror the entire other half vanishes. Is there a way to make both the halves independent so I can edit them differently?
I am assuming that by 'disable the x-axis mirror' you mean that you are unselecting the x-axis in the mirror modifier menu.
'Mirror' is a modifier; in order to keep any changes made by a modifier you will need to 'apply' that modifier: Once you have finished making the edits you needed to make with the mirror modifier, tab-out of edit mode, then click 'apply' on the mirror modifier. This will make the changes applied by the mirror modifier permanent. Then you can continue to edit the model as you like.