WARNING: The following contains spoilers for Season 1 of Hawkeye, now streaming on Disney+.
There are many secrets in the Marvel Cinematic Universe that superheroes create to keep their identities safe, but Clint Barton doesn't seem to make that big of an effort to keep his family's identity secure. In fact, the Hawkeye series completely contradicts Clint's wishes in Avengers: Age of Ultron to keep his family low-profile.
Clint's farm in Age of Ultron was used as a safe house for the Avengers, and every hero except for Natasha was surprised by the revelation that Clint has a wife and children. He even spends the first half of the movie lying about his phone calls to his wife, either just saying he has a girlfriend or denying the fact that he has one. When he does introduce his family to them, he further explains that Nick Fury helped set up the farm's security when he joined S.H.I.E.L.D. After the events of Captain America: Civil War, their new farm was exposed by government authorities and Clint was kept on house arrest for disobeying the Sokovia Accords.
Clearly, the government is aware of Clint's family after Civil War, but his loved ones become a potential target in Hawkeye. After discovering a Rolex watch was stolen at an Avengers compound clean-up auction, Clint mysteriously takes an interest in said watch. The end of the series reveals that it belongs to Laura Barton, who is a former S.H.I.E.L.D. agent, and that the watch could link to her identity. Somehow, Maya cracks the code on this since she comes into possession of the watch and has the names of Clint's family in a notebook, which creates a bigger mess for Clint to clean up.
But it's rather fanciful that Maya was able to track down the names of all three of Clint's children just from the watch in such a short amount of time. So how did she get their names exactly? It all may loop back around to Clint's holiday vacation in New York with his kids. Hawkeye Episode 1 sets up the New York background by showing Clint and his kids taking time to tour the city at Christmas and watch the popular Broadway show Rogers: The Musical. Even if Clint is the more low-profile Avenger, he is still widely known and fans or tourists are bound to take pictures of him and his kids.
Maya could've easily used the internet's resources to find these pictures and it wouldn't be hard to find the kids' names since they all use the same surname, Barton. And since Clint seems to want a relatively normal life for Cooper, Lila and Nathaniel, they presumably attend school and have friends, but they may use a fake surname in that situation. Nevertheless, their proximity to their father in one of the most populated cities in America during the Christmas holiday doesn't seem to bother Clint until it becomes an issue with the Tracksuit Mafia.
It's questionable how much of an effort Clint and Laura are putting in to protect their children from people who could have a vendetta against the Ronin or even Hawkeye. As a former vigilante and an Avenger, Clint constantly has unknown targets on his back. He has made it clear that his number one priority is his family and their protection, but doesn't go to extreme endeavors to execute this priority. But it seems after the events of Civil War that his valiant efforts may not be as necessary considering federal authorities are aware of his family, so he may be using that opportunity to allow his kids to have a more normal childhood with some Avenger privileges.
To see how Clint is managing his family's secret identity, stream Season 1 of Hawkeye on Disney+.