Switch to Profile view, click Edit Profile, and click the (probably shaded/barely visible) "Send an email" button underneath your profile picture. There are a number of privacy options here, although, notably, "anybody on the web" still means "anybody with a Google profile" (or at least I couldn't get it to work unless I was logged in).

Senders see a popup window and indeed G+ does not reveal your e-mail address, which is as it should be. Receivers get an e-mail entitled something like "Message via your Google Profile" with the sender's Gmail account's from name/address.

My comment on G+ encouraging the use of your "real" identity was just meant to highlight that apparently Twitter is the outlier as far as social networking sites go, with regard to whether you can have a completely anonymous (or pseudonymous) conversation via PM. With G+, it seems clear they just want you to use GMail.