First up, you'll need to be buddied with the other operator. Buddies are separate from queue assignments so that you can chat with anyone. Your administrator can use the Manage Contacts tool to do the buddying or you can add the other operator as a buddy manually in your staffing client. Once buddied, simply click the desired operator in your contacts list to start a private chat.
Notice that the chat management toolbar is missing since this chat is private and not over a queue. You still have access to canned messages, emojis, and an erase transcript function.