D.C. law defines the crime of "stalking" to include, among other things, "communicat[ing] to or about another individual" "on 2 or more occasions" in a way that intentionally, knowingly, or negligently causes a person to experience "significant mental suffering or distress." The law does provide that it "does not apply to constitutionally protected activity." But two D.C. Superior Court decisions in recent years, Gray v. Sobin (2014) and White v. Muller (2017), have read this exception as limited to speech on matters of "public concern"—speech on purely private matters, they have held, is not "constitutionally protected" for purposes of this law.
Say then that a woman in D.C. finds that her boyfriend is cheating on her and tells two mutual friends about it (on two separate days). If she should have known that a reasonable person would feel "significant mental ... distress" as a result of being revealed as a cheater (not unlikely, I would think), she's committing a crime.
Or say that her boyfriend had given her an STD, and she tells two mutual friends about that. She too is committing a crime, if she should have known that a reasonable person would feel significant mental distress as a result of some friends learning this. Saying such things actually isn't a tort: Though "disclosure of private facts" can be tortious, that tort, for all its flaws, recognizes that friends have to be free to speak candidly to each other, and doesn't punish disclosures to only a small circle of friends. But D.C. stalking law would make the speech criminal.
And of course this would even clearer as to other speech that is less justifiable but still generally protected. If a woman badmouths another woman twice—say, reveals that the other woman has had many sexual partners, in a social circle where that is still seen as shameful—that too can be a crime. Same if a woman tells two friends that she left her boyfriend because he was impotent, or just a bad lover, or too poor for her, and she should know that learning about such statements would cause him to feel "significant mental ... distress."
Now I don't know if there have been any criminal prosecutions on this theory. Instead, the cases that I've seen involve people seeking "civil protective orders" against the people who were saying distressing things about them; under such orders, the speaker could be ordered not to go to certain places, not to possess firearms, and of course not to continue the speech that the court found to be "stalking." But to get such an order, the plaintiff must show that the defendant had committed a crime against the plaintiff; so the cases I cite expressly considered what constitutes criminal stalking, and held that repeated private-concern speech that negligently, knowingly, or intentionally causes significant mental distress is a crime in D.C.
And of course the presence of such a law can deter speech even if there's no criminal prosecution. The protective orders, once issued, are backed by the threat of criminal punishment. But even without a protective order proceeding, a police officer or a prosecutor can often get results just by threatening prosecution. Perhaps the police will use the statute only in narrow circumstances; but, as the Court said in U.S. v. Stevens (2010), "The First Amendment protects against the Government; it does not leave us at the mercy of noblesse oblige. We would not uphold an unconstitutional statute merely because the Government promised to use it responsibly."
If you think that there should be a "private concern speech" exception to the First Amendment, applicable even in criminal cases, then you might support this statute. And Snyder v. Phelps (2011) did suggest that civil liability for private concern speech should sometimes be allowed, though there it was focusing only on speech that was "outrageous," that inflicted "severe" emotional distress, and that did so purposefully or recklessly, not just negligently—elements absent in the D.C. statute.
But I think the better view (especially for criminal cases, where the speech restrictions can't be defined as vaguely as they sometimes are in civil tort liability) is the one expressed in Stevens: "Most of what we say to one another lacks 'religious, political, scientific, educational, journalistic, historical, or artistic value"'(let alone serious value), but it is still sheltered from government regulation." Criminal punishment for any two incidents of speech about a person can't be permissible, even if the speaker should know that a reasonable person would find the speech to be significantly distressing.