This is an interesting perspective. Do you use a home-rolled email client/server, and only exchange emails with other people with a similar setup? Completely avoiding "third party apps" with respect to email seems like it would take quite a bit of effort.
I work in a startup that does zero knowledge email, so I think about stuff like this a lot. Would it be possible to do the work at "runtime" i.e. process the emails on the user's device?
I highly doubt it. I'd imagine the training data is several gigabytes if not terrabytes of corrospondance.
I don't know much about this type of AI work, but I can tell you a little about the scale of things where I do work.
I work with kana/kanji conversion and it is a similar problem as just getting the phonetics part of speech to text sorted out. First you convert the sound signal to phonetics, then you need to convert the phonetics to actual sentences. A surprising amount of English sentences have the same phonetics, so it is a non-trivial task to make sense of these. Try saying "Recognize Speech" and "Wreck a nice beach" out loud.
I'd dare say Japanese is even worse, so a lot of data is used to infer what the writer means with the phonetic input in kana/kanji conversion.
Baidu uses 2.5 terrabyte[1] of data for their kana/kanji conversion system, which has both an online and offline part, where the online part makes use of all the data and boosts performance considerably.