Hmmm, that’s strange. I’ve been unable to replicate this end. Did anything else happen around the time you noticed this, like updating WordPress or plugins? If so deactivate any third party plugins to see if they are conflicting with the Twitter api.
If you remove the widget from the second sidebar, does the first one resolve? If you put it somewhere else, and remove all other instances of the widget, like add it to the footer, does it resolve?
Is your mobile phone number in your twitter profile? This seems to be a new addition to their auth for the application set up for the auth codes.
Do you have any form of caching active on your site?