Orlando (CNN) -- Judge Belvin Perry, who oversaw the Casey Anthony murder trial, did not immediately rule Thursday on whether he will release the names of the jurors.
The names will not be released Friday, he said, and he indicated that he will consider setting a "cooling-off period" after which the jurors' names might be released.
A few news organizations petitioned for the names to be released, arguing the release is in the public interest.
In an extensive hearing, Perry asked the media organizations' attorneys a series of questions surrounding a central idea: whether potential threats to the jurors' safety and the attention they would receive from the media outweigh any public interest in having the names released.
Attorney Rachel Fugate, representing the Orlando Sentinel and The Associated Press, argued that juror information is "presumptively" available to the public. Perry then asked about cases that are exceptions.
One of the jurors has "very serious security concerns," Perry said. People had reached out to the juror's relatives even before the juror left the courthouse after delivering the verdict Tuesday, and even though the juror's name had not been released, Perry said.
"It is no big secret that some people disagree with their verdict. And some people would like to take something out on them," Perry said. He asked, "Is the court totally powerless when it comes to folks' safety?"
He also used himself as an example of how media fascination can affect people involved in the trial, citing something that had happened earlier in the day. "I can't even go to lunch now across the street" without being surrounded by cameras, he said, arguing that media have "sprouted up like weeds."
Fugate agreed that there are such concerns. But there are "beneficial effects" of releasing jury information, she said. "It enhances understanding," she said. The more people know, the greater the "public knowledge and understanding of the process and the verdict."
"If you have an anonymous verdict, it's going to cut against that verdict," and the verdict can be "delegitimized" in the public view, she argued.
Fugate and another attorney, Allison Steele of the St. Petersburg Times, cited previous case law showing that juror names can be kept private if there are specific, identifiable threats to individual jurors.
The media attorneys argued that any juror facing any threat should be protected whether or not his or her name is released.
And, the attorneys argued, others involved in the case, including the attorneys and witnesses, could face threats, and don't have the option of anonymity.
But Perry noted that jurors don't have a choice as to whether to participate. They receive a summons and are required to show up, he said.
Perry noted that jurors may choose to come forward. For example, one already gave an interview to ABC.
But he noted that most have indicated they want to remain anonymous. He asked, "Do we now ask the citizens of this county to add the burden of having 24-hour protection to those individuals" who may face threats?
Perry did not indicate when he might announce a decision.
CNN's Josh Levs and HLN's Adam Blank, Grace Wong, and Michael Christian contributed to this report.