Hi,
I don't know about all pigeons, I only have one, Sophie, who is a rescued Utility King Pigeon. She lives in the house and has freedom to roam as she sees fit. She lives primarily in a converted catbed "nest" on my desk. When she is sitting on eggs (either hers or fake), she stays in the nest until she wishes to go potty, stretch her legs, get something to eat, etc. Then she stands up and waits for me to put my hand out and steps onto it. I then take her into the kitchen where she will poop. About 90% of the time, she walks over to the sink and poops into the sink. The other 10% she goes on the counter and I catch it with a paper towel. If she is walking around the house, she may poop in the hallway, another room, anywhere but my office, where her nest is. In over three years, she has never pooped in that room anywhere.
During the week or so she is off her eggs, Sophie wears her PGWear from first thing in the morning until she goes to sleep in the evening. I change every couple of hours to a fresh one, even though the liner takes care of the mess. She does not mind the PGWear at all, in fact I think she sort of likes it. IMHO, the PGWear makes all the difference for a "house pigeon" because one does not need to follow around the pigeon with paper towels or worry that there will be an "incident" on bed, sofa, etc.
Hope this helps.
Best,
Bill B.