Most all .22 "bird shot" shells I've used are very short range, (these are the ones that look like a regular 22 shell case with the end crimped together). When you take one apart, the "shot" is not small round pellets, it's like fine lead granules.
In noise sensitive areas, I'm aware of a couple of choices, a .22 rifle with suppressor and sub-sonic ammo - very effective, very quiet, but expensive and a hassle to acquire the suppressor. Then there's a PCP (Pre Charged Pneumatic) air rifle. I have two Benjamin Marauder PCP's, one in .22 and the other in .25 cal. The advantage is many PCP guns come with built in "moderator" (essentially a non-removable suppressor). Since the gun is not a firearm, there's no $200 tax stamp and attendant hassle and wait as with buying a firearm suppressor.
The Marauder .25 PCP is very effective at taking squirrels - one shot one kill out to 50 yards (probably more, but that's as far as most of my shots). It's almost as good as my CZ 452 American .22lr with suppressor. The CZ is a little quieter using CCI segmented hollow point sub-sonics, but not by much. Both options are back yard quiet. Shot placement in important with either, though a little less so with the 22lr. I have rock walls in the back yard, so the ground squirrels have to be DRT or they can escape into the wall.
Two downsides to the Benjamin Marauder PCP. It's a little heavier than a typical light weight .22lr, and it can be a bit of a workout charging the air reservoir with a hand pump. I have a small high pressure electric compressor, but that's another expense, but then the ammo for the pellet gun is dirt cheap by comparison.