Can you embarrass someone privately if you feel that not saying something may cause that person public embarrassment, i.e., s/he may be embarrassed by multiple people? The key word here is MAY, but you calculate that there is a high likelihood that public embarrassment would occur but, there's no guarantee.
As an example - You notice that a man has his zipper open and part of his privates are visible. Telling him this will embarrass him, but if you don't tell him, and he walks in the street, the public will embarrass him, most likely. Of course, there is a possibility that the man may close his zipper before going out, or even if others in the public will see what's going on, no one outside may mention anything, in which case no one publicly would embarrass him at all.