We did our coop with interior nest boxes. The coop itself has exterior siding, and then also h as an interior wall (except it's plywood instead of say drywall like the inside of a house). My next boxes are kind of like a dresser within the coop- the front is open to the main part of the coop but the backs are closed (except when I open them for eggs) - so that's 3 layers of plywood that kind of insulate/dampen the noise. I don't have any actual insulation and my windows are open almost all the time. Definitely don't sacrifice any form of ventilation for noise dampening, especially in AZ.
Your best bet will actually be FANS. This will both dampen the noise and help keep the chickens cool. I've linked to a poultry exhaust fan that is made to take the abuse (dust!) created by the chickens - and it will of course help dampen the noise of the girls while keeping the air moving. If you point the exhaust fan so it blows into their run area, they get a nice breeze outside too!
https://www.valleyvet.com/ct_detail.html?pgguid=D74B298D-3C6F-4287-9F9F-4EA09B8D0A73