It is about time. Twitter has updated its app on Windows Phone with Notifications option. Twitter app on Windows Phone is one of less used app on Windows Phone due to the fact that it does not have notifications.
With the latest update (to version 1.2.4555.39631), it now has a Notification column on the settings. You can set notification for Mentions, Favorited, Retweets, Direct Messages and New Followers.
Not sure if it also includes Live Tiles or Toast or not (no options). It is a good start. Hopefully, it can become better. BTW, I am using ROWI (paid app) and Mehdoh (free app) on my Windows Phone.