As a pilot I really couldn't care less about where the frequency log, and the messages I've sent get shown.
As you say, Magnus, "otherwise violate the legal rights of others, including expectations to the rights of privacy and publicity", I don't think it's considered to be a violation of the code of conduct to show the Euroscope chat box window and/or Pilot client chat window, as pilots may be TEXT on VATSIM. I do have the expectation that my conversation with ATC, if I am a pilot is not streamed, I do believe that to be between the ATC and my self as pilot.
I've seen countless streamers (including the "big" profiles - some of them SUP rated themselves) stream frequency logs both from pilot and ATC clients without getting shot down by staff.
I would, however, be careful and make sure not to publicly broadcast messages considered "private" - e.g.
Individuals violating this rule are VATSIM Code of Conduct subject to the procedures set forth in Article I of the Code of Regulations which could result in removal from VATSIM and revocation of their certificate number.
A12§.)Members are prohibited from using online chat boxes, ATC text channels, voice ATC channels or the official internet forums, newsgroups and mailing lists utilized by and its officially recognized regions, divisions, ARTCCs, ACCs and FIRs to threaten, harass, abuse, intimidate or stalk others or to otherwise violate the legal rights of others, including expectations to the rights of privacy and publicity.
The pilot with "NOOB" in their scratchpad could have their mate on APP and thus find out anyway, doesn't have to go through a broadcasting platform to have a negative impact.