I have a small bathroom I've gutted, and on the other side of the wall from the shower is my living room. I'm thinking it might be good to insulate that wall for sound so showering (and toileting) isn't too loud from the living room. Any suggestions on materials to use?
Spray foam will fill the wall cavity better than anything else, and provide some mass to dampen vibrations in the entire wall. But if there's any wiring or plumbing in the wall, it might be advisable to run them inside conduit so they can be fished through later without disturbing the insulation. And it would make removing drywall a pain unless you put a layer of something between the drywall and foam.
Heh. 
This stuff.
I used this when I gutted and remodeled our master bedroom because of my very demanding work schedule at the time. I needed a quiet bedroom at all times of the day since I never knew when I'd be trying to get some sleep. It works. Don't forget a good solid core door also.
I just flush it's going to be a loud 'un. Try to pinch it closed while the water is still able to mask the sound, wait what I think is an appropriate amount of time and flush again.
In reply to Hungary Bill :
Timing is everything!
Some sounds were just meant to be shared!
I came to this thread looking for taco recipes.
Soundproofing drywall exists, or at least it does in Canada. IIRC, it installs like the regular stuff but is quite effective. It obviously costs more, but for the amount you need for a bathroom it wouldn't be a big hit to the budget.
I'll recommend Homasote 440.
If that’s a serious question, consider me impressed.
It's a serious question. I have the walls open anyway, so now seems like a good time to think about small problems like this. My shower and toilet are on one side of the wall, my couch in the living room is on the other.
I suspect just making sure that all gaps are well sealed and insulating with any common insulating material would be enough to make it pretty quiet. But the recycled Denim insulation (Ultratouch) claims 30% better sound dampening compared to fiberglass batts. It's also easier to handle but is a good bit more expensive.
The sound dampening drywall is neat - it has a layer of vinyl between two layers of gypsum on one sheet of drywall - but expensive. We built a "quiet room" for a tenant here at work using that stuff on both sides of the wall and traditional fiberglass insulation inside the wall cavities, and I was kind of disappointed in the performance. It's possible something was designed wrong (we followed an architect's specs) or that it's really quieter than the same room would be with traditional drywall, but it just didn't impress me as far as the sound performance, especially considering the cost.
I'd say, spray foam around any penetrations through the wall, caulk any gaps well, use the denim insulation, and make sure the door is solid core. That would get you a pretty acoustically separated space.
One step further would be to add weather stripping around the door. It would be a fair amount of work to do right but it can make an enormous difference in sound travel through the door. You can put a rabbet in the door stop to allow slip-fit (kerf) vinyl covered foam weatherstripping like this: https://www.homedepot.com/p/Frost-King-E-O-1-in-x-8-ft-White-Vinyl-Covered-Foam-Weatherstrip-Kerf-Door-Seal-DS8W-25/100115475 and it does a heck of a job. Just leave about a half inch gap between the door stop and the door. Obviously field measure this with the weatherstrip in hand - you don't want to get it all mounted up and have a gap or have the door too difficult to close!
A door sweep and threshold can also help a ton, but I think that's going too far as you then add a trip hazard entering the bathroom. And you can't use a sweep without a threshold or it'll drag on the floor making marks. Do make sure the gap under the door is as small as practical - don't have it drag the floor but keep it small - that will minimize sound travel.
mass loaded vinyl (MLV) sheets https://www.tmsoundproofing.com/How-to-Install-Mass-Loaded-Vinyl.html
foam is an absorber, not a barrier. you need density and thickness to block sound. rockwool mentioned above is good too.
I think you should install an active noise-cancellation system. That would be really cool. A microphone would instantly detect the reverberations of your flatulence and the system would respond by generating an "anti-fart," delivered by means of a large, powered subwoofer. Unsuspecting guests in the adjacent room would be blissfully unaware.
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