I write to the one I see - it could be email or regular mail. The therapist's response is not my goal. In fact, I dislike it when the woman responds.

I do not email the other one I see. I have never had an urge to do so. I rarely call her even though she encourages it. I do not find it especially useful for me to talk to her. Sometimes I call and just leave a message and add not to call me back.

For me, it is usually to get something away from me, not to engage in an exchange. That is why, for me, I do not see a problem with a person sending something to a therapist. They can put it in a file or whatever, but they really do not have to respond. They can even put an automatic response back like some do with phone calls "if this is an emergency - call 911 or go to a hospital"
I actually like sending letters better than email as it has no sense of immediate response.

The idea of a therapist "scolding" anyone has me much more bothered than what they do or do not do with an email. The therapist's boundary, it seems to me, is in not responding. A client merely emailing is difficult for me to see as any sort of boundary violation unless they hunted down a personal email address and began using that. The therapist controls whether or not to respond. As long as the therapist has made their ability/practice to respond or not to respond clear, then I really do not see all the fuss about boundaries.