The woodpigeon is a clean bird generally but them that we call town pigeons or feral pigeons are a health risk.Pigeons are a curse, at the natural gas compressor station where I worked for the last twenty+ years. If not checked, they'll cover the floors and the equipment in deep layers of crap. And the crap can carry a deadly fungus that's killed at least one person I know of...
A couple of years ago I carried a .22 cal., top-break pellet gun into the #1 compressor building, and over the next three or four days carried twenty seven of the fat feathered rats out in plastic bags. All done very discreetly after regular business hours, of course. The next year there weren't nearly as many, because there wasn't a whole generation returning home to nest.
And yes, they were tasty. Think pigeons in gravy... They're just oversized doves, and out in the country they eat what the doves eat.
I've done a lot of shooting in farm buildings of feral type pigeons.
They are derivatives of the rock pigeon that live on coastal cliffs or some are escaped racing pigeons.
An air rifle and flashlight at night is best for that job.