As of now there is no way to private chat on Yedda. You can however send a private message by email to a person simply by clicking on their name. This will take you to that persons profile page, where you will see that persons profile and an option to send them a personal mesage. You will also see an option that allow you to add them as a contact.
Adding a person as a contact is just an easy way to keep up with the answers or questions that one may ask or respond to. It is not an option that allows chatting. Likewise the email option is used mainly for seeking advice from a member when you do not want to post your question to the general public. You will find that some members may respond this way, others opt not to respond by email.
You never know, Yedda may someday add a chat box that those logged in could use to chat back and forth. I think it would be a nice option.
Good luck and best wishes.
By the way you will receive a notice from yedda that you are now added as one of my contacts.