The public has no expectation to privacy.
That means whatever activity you engage in the "public arena" from protesting, walking your dog, loitering, selling hot dogs, sidewalk shopping and even enforcing the law grants no protection based on privacy.
But that's really not the issue. The issue is that the police are accountable to the public and that should not be forgotten. Individuals video taping activities by the police do so out of protection of their own rights. And they do so based on a well established precedent of corrupt officers getting away with breaking the law. Imagine if Kathryn Johnston had home surveillance showing the actions by the police rather than the public relying on a near miracle where an informant refused to lie to cover up for the police. I also recall a woman who secretly turned on a tape recorder when police showed up her to home to interrogate her husband. They sent her away but thankfully the audiotape substantiated the physical evidence on the husband and resulted in a few bad cops being fired. One of them jailed if I recall correctly.
Private business owners already hold the right to videotape employees on the job except in those certain areas such as changing stations and restrooms. They do so legitimately to protect their business, the clients as well as the employees. Dashboard cameras exist on cruisers to aid the cops. Why can't a private citizen hold the right to exercise the same protection.
Yes, there are definitely some police activities which require discretion. But are we really worried about a rash of videos outing undercover cops posing as hookers. The most worrisome would be those undercover officers being exposed who deal with gang units and drug dealers. But these should still not withhold a right from the public to use those tools available to them for their own protection. Thankfully some courts are recognizing this and striking down arrests based solely on an individual videotaping a police officer in public.
Transparency and oversight. These do not exist on a practical level. New York City claimed to institute an citizen's review board with increased oversight following the criminal activities of some of the NYPD. It failed. Because it was not accountable to the people.
The Failure of Civilian Oversight (Gotham Gazette, October 9, 2007)
Videotaping police may not be a step in building better police/community relations but that is the fault of the PD's in their respective localities. When areas such as Tulia and Hearne, Texas experience what they did why bother trying to build a better relationship with law enforcement without first trying to establish better protections against abuse.
The police are accountable to the community. If the state fails to enact measures to protect the individuals in our society than the individuals must exercise those tools available to protect themselves.
The greatest problem is the interpretation of video evidence. That will exist but is no excuse to countermand it's practice. It is still far more reliable in many cases as opposed to eyewitness testimony. It is better to have some physical evidence than just two competing versions of a story. As in all things, a well educated public would serve to do better. So when a video such as the one above showing the arrest of Amy Goodman in which we can only see a couple of seconds where a police officer is pushing her away and giving her an order not to move past the line of police until another officer gives the command to arrest her. We see basically nothing prior. There is no grounds to measure any grievance wrongdoing by the officers. Yet, given the comments on the video you would think they are jackbooted thugs. So there will exist a mentality that refuses to critically examine the evidence. But that is still no reason to remove from the public a valid tool of protection against police misconduct.
I also see no reason why the majority of police officers would not welcome this.