No, because it can't run in the background. No application using the official SDK can, because Apple doesn't want background applications. And if they don't use the official SDK, they can't be sold on the AppStore.
That's actually one of the reasons for the price. To work around the SDK limitation, Beejive had to setup a centralized server. The server handles you IM sessions - it's the one that is logged for 24 hours. That's the server using push email to send notifications - since that's the only notification mechanism that works on the current SDK. The iPhone application merely connects to Beejive server.
In order for that to work, Beejive has to maintain a server that can handle potentially dozens of thousands of simultaneous sessions, each supporting several IM protocoles. That costs money, both in server maintainance and in design and development time...
I'm trying to understand this - what you say makes complete sense, but when(if?) push is enabled, wouldn't they still need some sort of centralized server? I mean, the push wouldn't be coming direct from AOL/Yahoo/MSN/Google, etc - wouldn't the push come from the Beejive server?