They are very intelligent little birds, despite the rep. people have given them. Pigeons and doves both will see you as their mate after a while, as many birds do, and will have a strong bond to you. If you have a hen, she may lay some eggs for you, which I'm sure they'd understand when you have to decline the invitation to share nesting duty, LOL. And males usually become cocky and dance and sing. Some people have said their pigeons will follow behind them (walking, not flying), and have even hopped up on their feet and untied their shoes before, haha. They're quite entertaining! But one thing to keep in mind, is they all have their own invidiual personalities, so not all pigeons/doves will acts the same as pets. Some are more outgoing, others more shy. And then some may give you a wing slap or two and play tug-o-war with their beak vs. you. And others may just prefer to cuddle up and play nice.
The only con I can think of really, is the droppings. But that's why God made BirdWear!...Okay, maybe not God, but a woman named Bonnie did!
Lovely pigeon, dove, and chicken sized diapers, in some fabrics that will surely set fashion statements in the feathered world!
One more thing. Getting a pair of pet pigeons will keep them from being quite as lovey-dovey on you. But both can be tamed and bonded with of course. They just don't devote all their love and attention to you in that case.