When you gain the power that comes with being a police officer, then everything you do when on duty and filling that official capacity should be open to public scrutiny. Since abuses of that power have very direct and damaging effects on the victims of said abuses, I feel that all officers must sacrifice some of their freedom, while on the job, in order to help protect civilian rights. Those who are good officers that go by the book in every situation (outside of emergency situations that can and do occur, that require nearly instantaneous reaction time) and don't abuse their power have nothing to worry about - if something bad happens that is questionable, having a record of it would also protect the officer.
Yes, I realize that's dangerously close to "if you're not doing anything wrong, you have nothing to hide". However, there is a difference between applying that to civilian individuals (including off-duty police officers) who have a very real right to privacy, and applying it to people who we have allowed to have legal power above and beyond what we allow civilians to do (for example, while civilians in many areas have the right to make a "citizen's arrest", they most likely do not have the right to use physical force to detain that individual, whereas the police do have that right). With power comes the responsibility to ensure that those powers are used properly, and having such interactions recorded as a matter of law protects the civilian and the officer.
I personally think this would be a good idea with civilians too, but the difference is, with officers (and other public officials with power above an ordinary civilian), those recording devices should be mandated by law and such footage available for review by any member of the public, but with civilians, the control over such devices (and indeed, the entire decision to wear them at all) should rest with the individual device owner. I really wish people would get over their Google Glass hangups and realize that, while in public, there is no right not to be recorded. Having a record of the day's interactions (that would automatically fade into the ether after, say, 24 hours, unless a segment is explicitly reviewed by the user and saved) would help prevent or at least reduce the "he said, she said" arguments that sometimes occur when a claim is made, because there is an impartial record of what happened (and in fact, in a society that doesn't have such a hangup about "public privacy", all parties would be recording, making for a way to corroborate what happened and have a fighting chance at detecting editing or other tampering).