Their friends only get to see your real name afaik,
and except for the person sending the initial invite needing to know or guess the other person's email, the email of either party is never shown anywhere at any point.
You will only know if the email you tried to invite was someone's real battle.net email if they accept the invite.

These combined make me feel safe enough to be going to use it with real life friends (good ones) and select few really close online ones.