Facebook itself used to have this level of privacy protection. You could join a group and post on the wall and in the forums for that group privately. If your privacy settings were to disallow non-friends from seeing your profile, then it would just show your first name and a generic placeholder as your photo. Your name would be unclickable and therefore untraceable to you unless you had a unique first name.

Ironically, right before the big push to erode users' privacy, it seemed that they might have taken it a step further and replaced your name with "facebook user", removing even your first name (I was never sure if this was intentional or a bug, but in any case it didn't last long as the big privacy changes came soon after).