How can I contact users without letting them know my email address?

Choosing to hide your email address from other Launchpad users only affects its display on your personal home page (https://launchpad.net/people/+me). When you use the "contact this user" link on somebody else's home page, the generated email will appear to come from one of your confirmed email addresses. The form will warn you about this.

Launchpad does not contain a full fledged messaging system that would allow to track messages back to their senders and avoid abuse. This is the prerequisite for sending emails without disclosing your personal email address.