I think it's actually a good idea to train them to treat the internet as a space without privacy, because it really is. Not sure if reading their emails etc is the way to go about it though.
This is called "code switching" and is just something that happens (with all people, but particularly children) in multilingual environments. A former teacher of mine was writing her dissertation on it when the students she was studying went home for holidays and returned speaking only their native language.