The outbox has to exist somewhere. Instead of a central server like the group chat you could use some sort of distributed storage... but then some node(s) need to be online at all times to deliver asynchronous messages.
You dont need asynchronous messages to be stored centrally, but you d need both to be online at the same time at some point.
That s a feature I never saw in a chat app, they always have a central crap keeping messages defeating any kind of serverless claim.
Here are some protocols that use noise (specifically differentially private noise):
https://people.csail.mit.edu/nickolai/papers/tyagi-stadium-e...