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.
Great info, thanks. I didn't expect the message to be visible in Push notifications, it's obvious to me that this couldn't work - especially with client side encryption. Whether 'Secret Chats' are truly client side encrypted is another question (see the discussion with one of your siblings).
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.