As of the 6th January, 2010 you can feed the chickens within your neighbours coops, you may find a Mystery Egg within. Feeding chickens gives you + 10 XP (August 31, 2010) , 100 coins, and also increases their chance of getting a Mystery Egg when they harvest their coop.
If you find a Mystery Egg after feeding a neighbours coop one is placed in your inbox (given you have sufficent room), you may also share it on your feed. (May 14, 2010)
You may feed your neighbours' chickens once every 24 hours. You have a higher chance of finding a Mystery Egg if you feed your neighbours' coop daily.