How do you enter these so they are visible to "inside staff" (e.g. school nurses and Admin staff for emergency contact reasons) but unlisted to "the public" (e.g. a school directory)?

I had that same question recently. What I discovered is that you can select "no mail" under the attributes tab in the address(either student or individuals). The address/phone numbers will not print on the directory but it is still visable in EE. I have not investigated how it affects labels and other merge documents when "No mail" is selected. In addition, if you select this it is the whole thing.. address and phone number you cannot select parts of the address.

Caution! If you mark "no mail" the individuals will not be included in labels and envelopes -- and if you don't mark the "print a label for those with no address" you will not even know is missing from your labels & envelopes.