I agree with Birdbabe. My close friend has two pigeons. They are house pigeons and pretty much run her place.
She is cleaning up after them all the time, but they don't shred like parrots. Honestly they probably easier to allow free range of the house. She had a single male pigeon for several years before she brought in her 2nd rescue.
She gave her boy little things to stack and place around. Popsicle sticks, cotton swabs, toothpicks, basically an assortment of little things. He collects items in one spot, sometimes claiming items that aren't really bird safe like car keys. He ended up needing a toy for cuddling/sexual release. My friend bought him a pigeon sized grey plush toy. So that was fun for him. Now he has a girlfriend
Pigeons can forage, though they do not dig for food. They need their food in site and clearly identifiable, but you can encourage the bird to fly to different locations by placing small amounts of food around and changing the spots regularly. Don't conceal the food from sight.
Congratulations to her for getting a lovely companion. Pigeons are awesome! Very gregarious and sweet pets.
Most of the "wild" pigeons are actually feral domestic birds. Pigeons used to be a common meat bird, if you heard of
squab. I feel like we are responsible to care for them, since we (humans) are the ones who brought them all around the world just to let them go to fend for themselves.