The Gwibber on testing does not work well with twitter. However the one in sid does(2.91).
So here is a little how-to:
add the following line to the end of it:
sid main contrib non-free
it's a src repository so don't worry, it wont update you to sid. Make sure this is the only line with 'sid' in it(from the official repositories).
now run the follow:
$ sudo apt-get update
$ sudo apt-get build-dep gwibber
$ mkdir gwibber
$ cd gwibber
$ sudo apt-get -b source gwibber
$ sudo dpkg -i gwibber-service*
$ sudo dpkg -i gwibber_*_all.deb
$ sudo dpkg -i gwibber-themes_*
You are done now. Feel free to delete the created folder and its content(./gwibber/).
More info about backporting sid packages to testing: http://wooledge.org/~greg/sidfaq.html#10
Hope it works