I’m having the same issue. After creating the application and saving the consumer key/secret I get the error message “Could not recognize the response from Twitter” when clicking the Authorize new account button.
I tried disabling all other plugins but still the same error. Running WP 3.5.1 and TWP 2.5.3.
I currently have totally separate installations, tens of them. And since they all share exactly the same core files it is very frustrating and time consuming task to update them all if my customers have forgotten to do so.
ManageWP seems to be the solution to this direction, although it’s much more complex that I would need.