Twitter is internally testing “push notification” feature for iPhone App. It appears that users who’ve iOS 4.1 installed get the option to enable Push Notifications for Twitter. They don’t appear to be working yet, but the feature is definitely there.
Here’s Twitter’s statement on the matter:
We’ve been testing push notifications internally. When we launched Twitter for iPad, there was a configuration error that caused us to offer push messages to a small set of users. We’ve stopped sending push messages, but users may see an option to turn on push until we release an updated version of the app. So, push isn’t ready yet but we look forward to rolling this out soon.”