Hi, Ileaneb - Snap! I asked a similar question yesterday, with respect to the pros and cons of using my.* VS bit.ly*, ow.ly etc.
My impression from reading the documentation is that when you publicise your post to twitter using 'Publicize', WP will automatically supply a URL shortener for you. But you could also put in your own URL shortener, which you obtain from a 3rd part service, using the URL of your blog post as input.
But I haven't been able to figure out what (if any) tracking stats are provided for the WP URL shortner. Nobody has commented on this aspect of my question yet, and the documentation doesn't give any hints that I've been able to find. This is tipping me towards using 3rd party, although it adds an extra step to my process. I think I will ask Staff and see what happens....
all the best, Heather