Privacy concern: displaying emails

We got this feedback from a user today: "I've just noticed that when I click on anyone's name, their email address pops up. I'm not happy about this, as it's bad from a privacy point of view. I don't want my email address visible to anyone who looks.

Please at least give us the option to hide our email addresses in the settings."

Personally I agree that this is a valid privacy concern. However, there is a pragmatic reasoning behind displaying emails: it's useful a) when you want to direct-message someone, and b) when you want to invite an existing user to a new group.

I imagine this won't be an issue in the future: case a) will be fixed when we have an internal messaging system in place and case b) will be fixed when we implement Trello-style usernames.

What should we do in the meantime to patch the privacy hole without creating a UX regression? Is the user's suggestion (add a 'hide email' setting) workable?