Messages in secret chats are stored on server until downloaded by recipient. So there is no such problem like in Skype.
Push notifications also work fine there, except on iOS they don't contain any message data, just "You have a new message", probably because server doesn't know what's inside encrypted message. Although havent tried their android client.
What is the point for server side open source? You can never trust that this code is launched on the instance which answers to your client' requests.
If you don't trust the backend in some particular way, nothing can help you to trust cloud storage. You should better take a look at end-to-end encrypted chats [1]. In theory they don't allow server to get access to any user data, except the fact of sending messages.
Have you tried the app? I've moved my top-5 chats from WhatsApp just because of speed! The messages are sent really fast. But if you know any other messenger apps for iOS, which are just about so fast, I'll definitely give them a try.
Anyway yep, these marketing stuff is a bit too dubious, but what should they write?
* is rather fast, faster than some applications
* can self-destruct, but sometimes can not (Ha!)
* etc
That's not the thing that happens in real world nowadays.
I guess, that the problem is in encryption key exchange. Cloud chats are easy to synchronize between devices. Secret chats require unique encryption key for each chat.
Push notifications also work fine there, except on iOS they don't contain any message data, just "You have a new message", probably because server doesn't know what's inside encrypted message. Although havent tried their android client.