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.