exception 'TwitterException' with message 'Bad Request'
I actually had a lot of trouble setting the Twitter module up on my localhost for testing, so I just created a blog post about how to set that up, that includes some tips on doing this with localhost: http://drupalwoo.wordpress.com/2013/04/22/how-to-show-tweets-from-your-twitter-account-on-your-drupal-website/

Here are the couple of things that I think my be relevant for you to check:


If you’re testing on localhost, use the following urls for your website and callback URLs (don’t worry, these are just placeholders, and your drupal site will send the correct callback URL when it connects):

Website URL: http://127.0.0.1:8000/twitter/oauth
Callback URL: http://127.0.0.1:8000/twitter/oauth

If you’re still getting errors when testing with localhost, make sure the version of the module you’re using has the patch #5 on this issue queue: http://drupal.org/node/1924478#comment-7260928 (if you have an older version of the module, prior to the incorporation of the patch, get the latest stable/dev release that includes it and uninstall and install the module)