Twitter now allows it's user to use HTTPS by default by enabling it on the Setting page. This new features is announced on Twitter Blog. To turn on HTTPS, go to your settings and check the box next to “Always use HTTPS,” which is at the bottom of the page. That's it. No more tricky thing or third party addon required to make sure your data is sent encrypted to Twitter's server.
If you need HTTPS for mobile, you should update your bookmark to https://mobile.twitter.com as Twitter currently don't share Always use HTTPS option to mobile users. Perhaps they are still working on it, but just to be sure, please update your bookmark first.
I wish to see more and more websites offering HTTPS protocol by default instead of staying with the old, insecure HTTP protocol that has been used for ages.