There is little doubt that the first 'captive' pigeons would have been taken from the nest and hand reared, much as the Amerindians in South America still do with birds and animals.
Early on, in the Middle and Far East, and later in Europe by the Middle Ages, this had progressed to sometimes, very elaborate and sturdily built 'Dove Cotes', many of these were so well built they still stand today. One reason, at least in colder climates (and in hot climates for protection from the heat!) for such construction would have been that with such thick walls and the enevitable build-up over the year of pigeon droppings, by Winter the temperature within the house would have been such that all year round breeding would have been possible, thereby giving probably the only fresh meat supply during the Winter months.
The same would also be true of the practice of 'Lofting' pigeons in the roof space above the house, stable or barn, the heat generated by the humans, animals and stored hay/straw for the few over-wintered animals, would also give a temperature conducive to breeding.
I have practical knowledge of my own birds rearing successfuly - seven from eight young hatched and reared during a period here in Germany where the temperatures were never above minus 5°C and often minus 15°C at night with no extra heat or light, as I had not long moved in and had not intended my birds to breed anyway, but they insisted! A good indication of just why, the pigeon was and in some places still is, valued so highly!
From my father, I heard that in India, one village practice was to build a hut over a pit (obviously protected from water entering the pit!). The pigeons would nest in racks on the walls, in earthenware pots, so shaped as to make them difficult for snakes to enter (!) and when the pit was full of excrement, the hut was just moved to another pit! Result? A ready-made, well-fertilized plot of land!
The variations as to just 'how' pigeons have been kept must be as numerous as the people keeping them but all have resulted from the pigeon's adaptability and beauty.
Another story, how true it is I don't know, was that the last Mogul Emperor died from a broken neck due to falling through the floor of his pigeon loft! Apparently he was not greatly missed by his people, as he spent so much time with his pigeons he was a terrible ruler! Shame he didn't repair his loft floor!
Regards and Best Wishes,