I think the federal circuit court decisions recognizing a right to videorecord in public places—decisions that have so far dealt with recording police officers—are correct: A right to speak must include some right to gather the information needed to speak (what is often labeled the "right to gather news"), and recording what government officials do in public places is important to be able to speak credibly about it:

It is firmly established that the First Amendment's aegis extends further than the text's proscription on laws "abridging the freedom of speech, or of the press," and encompasses a range of conduct related to the gathering and dissemination of information. As the Supreme Court has observed, "the First Amendment goes beyond protection of the press and the self-expression of individuals to prohibit government from limiting the stock of information from which members of the public may draw." … "It is … well established that the Constitution protects the right to receive information and ideas." … An important corollary to this interest in protecting the stock of public information is that "[t]here is an undoubted right to gather news 'from any source by means within the law.'" The filming of government officials engaged in their duties in a public place, including police officers performing their responsibilities, fits comfortably within these principles. Gathering information about government officials in a form that can readily be disseminated to others serves a cardinal First Amendment interest in protecting and promoting "the free discussion of governmental affairs." Moreover, as the Court has noted, "[f]reedom of expression has particular significance with respect to government because `[i]t is here that the state has a special incentive to repress opposition and often wields a more effective power of suppression.'" This is particularly true of law enforcement officials, who are granted substantial discretion that may be misused to deprive individuals of their liberties. Ensuring the public's right to gather information about their officials not only aids in the uncovering of abuses, but also may have a salutary effect on the functioning of government more generally.

But courts haven't figured out how far this extends, especially when we get beyond recording the police. Here is an interesting 2017 opinion (People v. Rivas) from the New York intermediate appellate court; Rivas was convicted of fourth-degree stalking, which punishes anyone who "intentionally, and for no legitimate purpose, engages in a course of conduct directed at a specific person, and knows or reasonably should know that such conduct … is likely to cause reasonable fear of material harm to the physical health, safety or property of such person," and of first-degree harassment, which punishes anyone who "intentionally and repeatedly harasses another person by following such person in or about a public place or places or by engaging in a course of conduct or by repeatedly committing acts which places such person in reasonable fear of physical injury."

The [government alleges] that defendant resided in a building located on Burns Street and that, on or about November 9, 2010, while the complainant, a mail carrier employed by the United States Postal Service assigned to a mail delivery route in the Rego Park section of Queens, was on her mail route in front of that building, defendant approached her and accused her of not delivering defendant's mail, "stating in sum and substance, you could lose your job for that, you better be careful.'" The [government] further alleges that defendant followed and videotaped the complainant with a video camera on numerous occasions from November 9, 2010 until May 31, 2011, at various locations on her mail route, approximately three or four times per week. Defendant also allegedly posted on an internet website in April 2011 that the complainant was guilty of misconduct in her duties as a postal worker, and included several derogatory statements about the complainant's weight and behavior. On April 27, 2011, a postal inspector went to defendant's home and informed her to cease her conduct and refrain from harassing the complainant. Defendant agreed and told the inspector that she had removed the internet posting. However, defendant continued her activities, particularly on July 25, 2011, when defendant circled the block approximately five times while holding a video camera, and pointed it at the complainant, who was retrieving mail for her route. Defendant also made an obscene gesture toward the complainant. The [government] also alleges that, on August 1, 2011, defendant pointed a video camera at the complainant, approached the complainant and stated, in sum and substance, "oh it's her again she's stalking us." The [government] further alleges that, on August 8, 2011, while the complainant was in front of the Rego Park post office, she heard a car horn, and then observed defendant looking at her while driving a vehicle. Defendant then stopped in the middle of the road, stared at the complainant, and drove off. Moments later, defendant's vehicle came back and defendant again stared at the complainant. Defendant's activities continued until August 12, 2011, when she was arrested. The [government] alleges that defendant's "actions caused the complainant to fear for her personal safety and to fear losing her job," and "caused her annoyance and alarm." … On appeal, defendant contends [among other things] … that she was merely exercising her free speech rights, obtaining information about a public servant performing her official duties, and was merely traveling around her own neighborhood. Here, the extensive allegations of the factual part of the superseding information were sufficient to render it facially sufficient with respect to the charges of stalking in the fourth degree in violation of Penal Law § 120.45 (1) and harassment in the first degree. It is objectively reasonable that someone following and videotaping another person in a public place dozens of times during a 10-month period knows or should know that the other person would be afraid for her safety and fear physical injury. Defendant was doing more than videotaping a postal employee at work to obtain information, or walking and driving through her neighborhood, and expressing her freedom of speech. The allegations sufficiently established that defendant was engaged in a months' long course of conduct directed against a specific person….

A few possible reactions:

This shows that the First Amendment shouldn't protect videorecording in public places, if there are laws that forbid it. The First Amendment should protect videorecording of the police in public places, but not of postal workers, because (a) it's important that the public be able to monitor the behavior of the police but not that important that it be able to monitor the behavior of postal workers, (b) the police are likely to feel less threatened by videotaping, because the police are armed, or (c) a combination of both. The First Amendment should protect videorecording of any public servant (or perhaps any person) in a public place if it's done once or twice, but not if it's done very often. The First Amendment should protect videorecording of any public servant (or perhaps any person) in a public place if it's done in the place where the recorder happened to be, but not if the recorder is following the subject around. The First Amendment should protect all videorecording of any public servant (or perhaps any person) in a public place, though perhaps subject to narrow restrictions about, say, how closely the recorder can approach. Something else.

I'm not sure myself what the right answer is, and I'd love to hear what you folks think.