Hi, I couldn't find anything related to private messages, only to authentication. Have a look: http://ubuntuforums.org/showthread.php?t=1571574
Please else share a screenshot of the error. And how do you repeat the error to show? (I.e., what button if you press it always causes this error to pop up.)
Attached two screenshots. One shows the error in Gwibber itself (on top), and the other the message from Twitter about Gwibber not being able to send or receive private message when authenticated (scroll and follow red error on screen shot).
I do not press any button to get the error, I simply add my Twitter account. I have also tried remove Gwibber from Twitter and re add my account. Same problem. Other apps authorizes without this error.