Opening arguments in former President Donald J. Trump’s criminal trial will not start until next week, but media coverage of the proceedings has already been met with partisan furor.

Jury selection, which began on Monday, produced a full 12-person panel by Thursday and ended Friday with the selection of alternates. Over the course of the week, many conservatives claimed that the selection process had been rigged against Mr. Trump, while some progressives argued that the press was releasing too many details about possible jurors, putting them in potential danger.

Journalists and commentators covering the trial, being held in Manhattan, face a tricky balancing act: inform the public about one of the most consequential cases in recent memory while not putting its participants in harm’s way.

The judge, Juan M. Merchan, has ordered that the names of the potential and selected jurors remain confidential. The proceedings on Friday kicked off with a warning from a court official to reporters and members of the public attending the trial to follow courtroom rules around filming, photographs and technology.

Several news outlets, including The New York Times, have published some information about the lives of the people in the jury pool, such as their profession and the neighborhood where they live.

Jesse Watters, a prime-time host on Fox News, went further than many. On Tuesday, he showed footage of one prospective juror who had been excused. The woman explained that she had told the judge that she could be unbiased, then explained that “it’s very difficult for anyone really in this country to come to this without prior opinions.”