You don't need an account and you can encrypt the message before queuing it on the server. Silly stupidity inefficient and won't work offline but it is quite easy to make a generic push message scheduling server that just pushes around blobs as requested by client-side code.
That's an interesting point. It's still pretty messy and inefficient, but you're right, you could at least get away with not exposing user data, even if you're still transmitting it around.
That's a good suggestion, I should play around with that some time.
