Today, Mr. Pompano said, the district relies mostly on tips from students, a system that works well if there is an atmosphere of trust. “It goes back to human intelligence, where kids have at least one trusted adult,” he said, “knowing what they’re telling them is confidential.”

In a few cases, school administrators said, monitoring services have helped them identify students who appeared to be at risk of harming themselves. More rare were instances in which an imminent threat to others was thwarted. In 2015, as the first anniversary of a shooting at Florida State approached, a post expressing sympathy for the gunman and an intent to visit the campus was intercepted by Social Sentinel, the campus police chief said. The man was stopped on campus and warned to stay away. When he returned, he was arrested.

Patrick Larkin, an assistant superintendent in Burlington, Mass., said he receives alerts on his phone in real time from Social Sentinel. “Nineteen out of 20” come from people who are not even his students, he said earlier this year.

Real threats, administrators said, are more often flagged by vigilant users, as was the case with the Parkland gunman, whose troubling comments on YouTube were reported to the F.B.I.

Mr. Larkin said Social Sentinel helps him sleep easier at night. And because it can track only public posts — nothing that requires a “friend” request — he doesn’t see it as an intrusion.

“My concern was, what if it’s some odd hour and some kid tweets something I don’t see?” he said.

Mr. Margolis said it is hard to demonstrate that harm has been averted. “How do you measure the absence of something?” he said, adding that Social Sentinel’s algorithms have improved in recent months.

One client, Michael Sander, the superintendent of Franklin City Schools in Ohio, said he had planned to contact the police about a Twitter message that read, “There’s three seasons: summer, construction season and school shooting season.” But the poster appeared to attend school in Franklin, Wis. — not Ohio.