I had posted this question on a couple of other threads before I saw this one.
I don't have much experience yet with chickens as I just got mine a month ago. However, I did use to raise and race homing pigeons.
I would think that the pigeons would have more or less specific requirements that might not mesh well with the chickens in terms of their being in the exact same loft. For one thing, you don't want to use chicken wire in a pigeon coop because the birds will grab onto it and flap their wings against it, shreading their wing feathers. Dowels are usually used for that reason.
Second, pigeons don't really need much of an outside run if they are habituated to that loft. They need to exercise their wings by flying around, which is best done at set times just before feeding.
The rest of the time they only need a small inside area to nest and a cage that projects outside but is not on the ground for fresh air, which does not need to be large either. Again, it should be short so they don't want to climb the walls as you can't use the dowel approach outside.
There are also different feed requirements and there could be disease transmission issues.
If you must keep the pigeons in the same building as the chickens, I would build something for the pigeons up high that you can access by opening doors at chest or eye level, with an outside cage, also up high, opening to the opposite side as the chicken run. You could also have a coop with a central hallway for yourself and pigeons on one side and chickens on the other. You could construct the chicken nest boxes to have a flap to open into the hallway so you can access the eggs without having to go inside the coop. The feed and supplies can then be kept as well in this area.
If your pigeons can't be flown outside, you will need a much larger enclosed run which should be as long as possible so they can fly from one end to the other. This can be on the ground so you have an entrance to get in and have access to the birds.
Basically, pigeons are fliers, chickens are ground birds.