The Maskana of most Poskim seems to be that chatting is without a doubt Assur while sending the email isn’t because you’re not directly causing anything additional (i.e. they have to be checking their email to begin with anyway). This would then become Assur if the person has a smart phone which immediately alerts them to an incoming email.
Many Poskim are Machmir on both, though, whether M’ikar Hadin or as a practical reality.
R’ Schachter has a massive Chumra that there is an Issur (he might even hold D’oraisa) of having your Ma’aseh have a Totza’ah come from it on Shabbos. Thus, chatting with an open chat window in Israel (or an open email inbox) is an Issur on yourself, not just a Lifnei Iveir-type Issur.