The simple answer? Don't give the videos out to the public.
What you do is you compromise. You put a procedure in place where members of the public who are directly involved in a case, or their legal representation, can request copies of the necessary videos, perhaps by putting down the badge numbers of the police involved, case number, whatever. If the people or owners of the property being filmed consent, then it can be openly released to the entire public.
Because frankly something like 50%-80% of police work is BORING. It's sitting around in between movements. When something does happen, 90% of the time you aren't there when it actually happens, you just show up afterwards. You walk around, poke into shit (sometimes literally), talk to people, take notes. It's important, but not terribly impressive on the face of it, so most people will not be terribly interested in watching it.
Now in cases where the videos are actually important, there are two things going on: first, the police department has a vested interest in not looking bad. Secondly, the privacy of the citizens involved is a concern, as noted. There is no real need for the public to be aware of what happened unless it was a case of police misconduct or other illegal activities, OR the citizens involved are fine with publicizing the case.
The example above of someone's house being broken into is actually reasonable. Say they're into... kinky sex or something and have all kinds of interesting equipment in one room. They might be absolutely ordinary and upstanding citizens otherwise, they just like a little extra spice in their sex lives. Is it really necessary for something like that to be plastered across the hometown newspaper the next morning? So confine that video to the legal teams involved (say they want to sue the police department for not handling the case well or something). That way privacy is maintained, but so is transparency.
Now in cases of police misconduct, say a cop beats some hitch-hiker or something up by the side of the road. If it was filmed by the cop's body-camera, the hitchhiker really has no reason not to release that video to the public unless it shows them clearly giving the officer reason to rough them up (trying to fight the cop or something).
It's a strange world. Let's keep it that way.