iTwitter had push notifications for mentions and direct messages, but only from other iTwitter users. You've now got two options for real Twitter push notifications if you're dying for them: iTweetReply and Boxcar. Updated.
iTweetReply is a full-blown regular Twitter app for $2, though it's not the prettiest or most feature-packed—except for the push notifications for mentions and direct messages. Boxcar, on the other hand, works with the Twitter app you already use—when you get a push notification of a mention or DM, it'll open your favorite Twitter app. Interestingly, it costs more than iTweetReply, at $2.99.
Since DMs go to me as text messages, I don't need push enough rightthissecond to shell out another $3 on top of the money I've already shelled out Twitter apps. They'll get around to it. [iTweetReply (iTunes), Boxcar - Thanks Marsala]