This way, you might not have the full conversation on each device - e.g. you write "a" to alice on your phone, turn it off, turn your pc on and receive "what do

This way, you might not have the full conversation on each device - e.g. you write "a" to alice on your phone, turn it off, turn your pc on and receive "what do you mean?" from alice - the conversation is otherwise empty. That's not a good UX. You'd have to store the chat history somehow - and thus loose the privacy aspect of not storing it.