My only LEO contact at all was with the guy we spoke to about filing an accident report when my wife was the victim of a hit-and-run accident. He never asked to see either of our IDs (my wife was carrying at the time too), advised us to not file a police report about it, and wound up dealing only with the insurance company... ...and they don't know about our CHLs.
I am actually surprised by the number of people on this board who get stopped; and I'm not claiming to be a saint or anything. I do speed a bit... not recklessly or necessarily intentionally, but definitely over the posted limits with some degree of regularity. But I do rely on my radar-detector to warn me, and between that and just being vigilant I've avoided being stopped, even in sneaky speed-traps.