There are really two issues, the audio and the video.
Video is legal anywhere in the country as long as it's not somewhere that would be considered intensely private. Your home, a restroom or dressing room are examples of intensely private areas.
Audio is different in every state although the courts are beginning to combine the two where the recording media is video with an audio track.
In Virginia, in public places, if you can hear it, you can record it. If you use an electronic method of intercepting the conversation such as over the phone, you must be a party to or have the consent of a party to, the conversation.
Most Police are at least generally versed with the laws now because they do it too. They can get nasty sometimes and demand you turn it off. That's up to you, I never do...absolutely never.
I was videoing Skidmark and Grapeshot in Ashland with an Ashland cop who was putting on his best good guy act for the camera. After a while he looked at me and said "You have enough yet" I said "Why, you gonna do something you don't want on the news"?
A couple of months ago I videoed a Richmond Cop dragging an unconscious man (probably drunk) while they were waiting for the rescue squad. He got mad and demanded I turn it off. I just said "Not a chance"...Then he demanded ID. With the camera running I said No. He said "Oh yeah, why not?" My reply "Because I don't have to show ID and there isn't anything you can do about it".
He pulled out his phone and videoed me for a few seconds then stomped back to the fellow on the sidewalk.
A few weeks later I gave the video to a Lt. with the department who was real unhappy to see his officer dragging the man around.
To sum it up. There is no privacy in public, not for you, me or the cops.