For example, you probably wouldn't mind if I walked past you in the street and saw you go into a shop. You might object if I followed you around from the moment you left your home, filmed every shop you went into, what you bought, what you paid with (including any visible numbers on your cards, any PINs you typed in, what your signature looked like, etc.) and then put this on the web along with your name and address.
The idea that privacy is some sort of black-and-white concept is part of the problem.
You're right; I might. I might also press charges for "stalking", which many jurisdictions have termed that kind of behavior, making it specifically illegal.
In particular, is it possible that no individual act of recording personal data observed in public would constitute an unreasonable invasion of privacy (whatever we choose to call that in law) yet the combined total still has the effect of stalking? Because that is the real problem that, IMHO, the law hasn't caught up with yet. In a connected, database-driven world, you can't judge everything in isolation, because it is often the cumulative effect that is the dangerous thing.
Please read my full comment. It does mention that an officer would have to be in public. In other words, recording him in the mens room would not be appropriate.
There should be a distinction if the party being recorded is in public. Courts have routinely ruled to the effect that one has "no expectation of privacy" when out in public. That the party being recorded is a public servant, engaged in (purportedly) public service just makes it all the more ... well, public.