Tomatoes love the nitrogen as well. I have yard birds and no, nothing will grow where the birds have access to, they will eat it before you can see it. Also, a little nitrogen goes a long way so it is to be used sparingly. I wouldn't expect to see much growth around the coop, but you can make a slurry to control how much and how quickly the garden is exposed to the nitrogen. Find some nitrogen loving plants to throw around the coop to collect some of the excess from their area. I would go a few feet out as a little nitrogen goes a long way and can cause most plants a bad time when starting. Also try to get the nitrogen into the soil before you grown, do not add bird poop after you have planted, get it into the soil a few months before planting what you want to grow. There are other complementary growing methods you can use as well to help soak up excess nitrogen and put other nutrients into the soil.