Nope.
You'll notice that for the Professional listings, there is a green tab in the upper right corner of our profiles that says "Contact Me", which enables you to send a private message to one of the Pros you see on the site.
This button is NOT available on the Homeowner listings, ensuring your privacy, but not giving you a means to message another User privately. Perhaps if you send an e-mail to support@houzz.com and ask if there is a way around this, as the only other way I can think of is to tell someone your e-mail address in a Discussion thread (like the OLLD threads) and then delete your post after they see the e-mail (but then so will anyone else who happens to be reading that thread).