Go to any home center or insulation company that sells bags of cellulose.
Sound insulation bathroom walls.
Since sound can be transmitted through the air the material is efficient as it adds bulk and absorbs air and sound from the walls.
It may be possible to do the work from the bathroom side if there are no fixtures on the wall unless you want to work around or over them.
Your first thought of blown in insulation is a good one.
Fold up thick fluffy towels on your towel racks and cabinets.
Otherwise you may find it easier to work from the dining room side.
You can also find decorative wall panels wrapped in acoustic cloth designed to double as soundproofing and artwork.
Rockwool safe n sound 3 in.
To reduce echoes arrange the room so that two sections of flat wall face each other.
It provides excellent temperature and noise control and comes with a vapor retarder or without a vapor retarder for use in hot humid climates.
Add shelves racks or other accessories put up a painting or hang potted plants on the walls or from the ceiling.
This mineral wool insulation comes as six pieces and weighs 54 pounds.
These absorb noise and won t be damaged by normal bathroom heat or humidity.
The best way to soundproof a bathroom wall is to add an extra layer of drywall with a damping compound like green glue and a soundproofing material like mlv in between.
Additionally it s definitely not the cheapest method too.
Soundproofing stone wool insulation 12 bags.