Noise is nothing more than pressure waves in the air caused by something
moving. The further and more rapidly something moves, the louder the
noise it creates because the stronger the air pressure wave it creates
with each movement. You can prove this to yourself by setting off your
smoke alarm intentionally, and finding containers to keep it in to
reduce the noise. You will find that a pyrex casserole pot works really
well because pyrex is a dense heavy glass that is very strong and stiff.
It moves very little in response to the sound waves from the smoke
detector hitting it, and so the only noise you hear is the result of the
pyrex moving, and that's very little. The result is that you hear very
little noise with the smoke detector in the pyrex casserole dish because
the walls and top of the pyrex casserole dish are hardly moving at all.
That is, you can have a tempest in a tea pot, but if the walls of the
tea pot don't move, then no one can hear the tempest inside.
Cast iron vent stacks were the standard for decades. Cast iron is much
heavier than ABS or PVC, and so cast iron moves LESS when it vibrates as
a result of water plunging through it, and the smaller movement means a
quieter flush heard downstairs.
You can eliminate the noise if you can eliminate the movement of the
plastic Vent stack in your wall.
I would use something inexpensive like wall paper or masking tape to
cover the stud cavity where the vent stack is, and then fill that stud
cavity with expanding foam. That way, you can eliminate it completely
should you ever want to by simply removing most of it with a knife, and
then peeling the wallpaper or masking tape off.