Wow! I really appreciate the array of responses here.
Yeah, if it were a "try it and see" sort of thing -- and if they hadn't already proven to be so "our kids are perfect, how dare you" about noise -- I'd be open to it. I mean I grow a good deal of my food, get my electricity from my roof, I'm all about sustainability. But given the reactions so far about noise...I've asked three times for some quiet. The first time the guy was all "oh of course, very sorry", the second time he was the pandemic housedad and pretty strung out, and he got all bristly, agreed to keep two hours a week scream-free so that I could teach my online classes (my students could hear them through zoom), and then ignored his promise, also got very touchy about the idea that yeah, he's really audible on his phone. The last time he rage-pruned my apple tree. (Boughs went over his yard, and I'd said he could prune if he didn't want the apples or ask me to get the tree pruned, but I wasn't expecting this emotional thing with broken branches...it was a mess, I had to get an arborist in.) The wife went all white-noise about it and suggested essentially that I not use my office or bedroom, which are nearest their property. Eventually they did quiet the kids down but now they're super-chilly as neighbors, no hello, nothing. I've shrugged and decided I'd rather have frostiness than that much noise.
One of our city councilors recently adopted the neighbor veto as a cause along with wanting to get rid of the single-family home clause, so I suspect we'll see it come up. In theory I'm really not wild about neighbor vetoes about property use, and a lot of councilors weren't happy about it either when the rule went in. But from what you guys are saying, it sounds like it's pretty likely that yes, the chickens would just add to the noise. (Now if I could get them to swap the chicken noise for the kid/guy noise, I'd be seriously tempted. 0% chance, but what a nice fantasy.)
Maybe, if it comes up, council will be persuadable about lot size and distance from houses. I don't think we have any rules about that right now, but as I look around, I see a lot of towns requiring at least 6000 sqft, some wanting 0.25 or 0.5 acres, a few wanting 1 acre, 5 acres for a rooster. (I get that the one councilor sees SFHs an equity issue, but it's also an equity issue for a duplex or apartment neighbor next to noisy birds.) I think we'd also need a way for neighbors to object to small-lot chickens in a way that leads to permit nonrenewal and that doesn't require a bunch of neighbors to complain -- if a neighbor's health/household/livelihood is being affected by chicken noise or smells, and efforts to work things out aren't working, then the human neighbors take priority.
Also, I hadn't thought about how much room a pen and decent room would take up back there...good point about that. They do have a big playset in the backyard, but I think the guy is thinking cramped quarters. He's also thinking about keeping the chickens in their under-a-bedroom garage during our long frigid winters (!) which just didn't sound great to me, for them or for the birds -- but they'd be living with it, not me.
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So I guess I have a followup: you guys know about chickens, what do you think is a decent amount of space for chickens? Like how big do you think a pen should be, per chicken, to keep them happy, and...say you had someone at home who needed quiet, how far away would you want the coop to be from your house so that you weren't hearing the birds early in the morning or through a good chunk of the day? (A lady on youtube was kind enough to record her chickens -- she's got a good big suburban lot, and from 100' away, it sounded like no problem at all. As she got closer, though, wow, those birds were loud, and they weren't even making laying noises, just nattering at each other. She probably had a dozen birds in there.)
Thanks again, and I really appreciate this.