Per The Iconfactory's blog:
We’re pleased to announce that Twitterrific 5.2 is now available in the App Store and brings with it our most requested new feature – native push notifications. Once enabled in settings, Twitterrific will now notify you when you receive mentions, direct messages, new followers and more. We’ve even added an option to display unique icons along with the messages so they are easy to discern on your device’s lock screen.
Please note that Twitterrific’s new push notification service is in Beta test and will be rolling out gradually to groups of 1,000 users over the next few weeks. If you cannot enable push notifications on your first attempt, just try again in a few days.
As of this writing, push is still unavailable to me. Hopefully I gain access soon.