While there seems to be no shortage of Twitter clients on the Google Play store, they all seem to be a mixed bag of different features and reliability. Some will do push, while others are free; some won't do notifications, and some animations are downright sluggish. Sadly, unless you have the time to go through every app to see which one fits the best, you'll likely need to settle for "close enough."
Echofon looks to add one more to the pile with the beta of their popular client. As a user of Echofon on iOS, I can say that they've definitely crafted something good over there, but at the moment their foray into Android seems a bit perplexing.