Bird droppings can be dangerous to humans and carry dangerous pathogens. Seriously, watch "House" or "Dirty Jobs" sometime and you be sitting out on your porch in your underware with a high powered beebee gun with a scope. If you just want to use that area, I would shovel up the top inch or two of dirt and then put some good potting soil on top. You can put some manure on top wether it's from a bag or cows or horses it's fine. After that, just till the dirt up to mix it and plant your seeds. They will grow pretty fast in this environment.

You probably want to compost the pigeon droppings first before using them. I am not sure about pigeon droppings, but I know chicken poo can burn plants if not cooked (compost) first. If you use the hot compost method, most pathogens should be killed. Mix some straws and dirt with the pigeon poo and keep the pile moist and covered. It should heat up and decompose in a few weeks, then you should have some pretty good compost to start your veggie garden with… Good luck and happy gardening!

I don't see any problem with using the area. I think it would be a good idea to get a soil test for the pH. It's probably going to need some big time lime. But you need to test the soil to be sure.
Nitrogen leaches out of the soil very fast so it could actually need fertilizer.

I'd be worried more about fungal lung diseases like histoplasmosis.

At least wear protective masks while you take soil samples for nutrient analysis, or do anything else that disturbs the soil.