Within Teamwork Chat, you can easily start a conversation with someone from within your Teamwork Projects site.
On your Projects site, go to the site-level People tab and select the user you want to interact with. At the top of their profile, to the right of their name, you will see an option to 'Start Chat':
Once you click the 'Start Chat' button, it will automatically launch the web version of Teamwork Chat in a new tab, opening a conversation thread with the selected user.
For more information, see: Creating a conversation or channel