It's disingenuous to say we, as the public, have no interest in what the police do while discharging their duties. It is a public office, after all.

And I think the activities of the police should be scrutinized heavily because they have an incredible amount of power and are accorded a very wide latitude. The police are the only people who can legally shoot you and not be tried. They can arrest you and throw you in jail. They can charge you with crimes, invade your house, and smear your reputation with the public. A police officer with a vendetta can absolutely ruin your life if not outright take it from you.

So yes, I think it should be okay for people to video the police going about their business.