Hi Sam,
you won't get many with your .22. You'll need a shotgun to make it worthwhile (you'll be amazed at how many you see within shooting distance when you're unarmed but as soon as you have a gun in your hand they won't come anywhere near you!!).
I love pigeon, it must be one of the most underated meats. Wizard is spot on - only the breasts are worth eating, although the French love to cook the whole carcass in the pot but they're very time consuming to pluck - I shoot them by the bucket load, if you do hang them, which is not really necessary, you'll need to ensure you empty their crop which is more often than not full of rape leaves at this time of the year & can taint the meat if left. To breast them, put them on their back & pull their legs apart so that the skin tears, then simply pull the skin & feathers away to expose the brest meat as wizard suggests, then with a knife, slice away the breast meat either side of the breast bone & you'll have two delicous plumb pigeon breasts ready to cook or freeze.
Try slicing it & using it instead of beef for 'pigeon stroganoff' or lightly fried with black pudding. Mmm - I'm making myself hungry

Regards