They don't seem to carry any more in the way of disease (etc) than any other animals, according to the New York City Dept of Health and Mental Hygiene [
http://www.nyc.gov/html/doh/html/epi/epi-pigeon.shtml ]
They say that diseases that humans can get from them (actually, their droppings) is mostly histoplasmosis, cryptococcosis, and psittacosis, and there aren't many cases of them.
I've never heard of them coming down with Mad Cow Disease (BSE) or that 'hot' E-Coli 0157:H7, both of which show up in animals that we deliberately pay to eat. They're not even free.
And I'm not sure, but I think pigeons are mostly dark meat.
Just keep in mind that people ate passenger pigeons right into extinction in slightly less than 100 years.
Sue