Hi. I joined up to thank people for this discussion, and offer a few of my own experiences. It's nice to know that this was a well known issue!
I was finding broken eggs in the coop and production besides was way down. I assumed one of my girls was the culprit but couldn't see any of the classic signs that I remember from childhood: yolk on beak, same chook repeatedly sneaking back into coop etc. Then I happened to look out and see a carrion crow wander quite casually up the ramp and in through the pop hole. I would never have imagined that it would go indoors to steal eggs! Such a bold creature, smart too, you have to admire it.
I was already having problems with the same crow because I'd seen it stealing the chickens feed. How can one crow eat as much as 5 girls combined? - and then want my eggs too?
It so happens that I have a 13x13ft pen around my coop. When I first started keeping chooks they were totally free range. But, after losing one to a early morning dog walker I decided it was better to build the pen and keep them there 'til about mid day. Beside keeping them safe it also encourages them to eat the proper layers meal (boring, but good for them!) , and lay in the proper nest boxes.
Well, it turned out to be relatively easy to build a "tent" of bird netting attached to the top of the pen. With the pen gate closed as it is until midday, the crow simply can't get in - and of course I collect the eggs when I let the girls out. It's only been a couple of days, but egg production has suddenly shot back up, and feed consumption way down, so it's looking good! Quite cheap too.
This isn't the first time I've had a problem with a carrion crow, and I know from experience that bagging it with an air rifle is easier said than done. They're very smart, and with the female acting as lookout it's hard to surprise them, which is the only way you'll get close enough for a shot.
p.s. I'm in the UK. Carrion crows here are larger, smarter than other crows, and form bonded pairs (not flocks). They are very territorial.