Two different acoustic engineers have told me that ordinary fiberglass thermal insulation batts with no facing will perform as well as anything you could put in yourself.

The expensive stuff is mineral wool (Ruxul Safe 'n Sound at Home Depot) but it won't make a noticeable difference IMO.

Insulation in a wall is not a sound barrier; all it can do is absorb a small amount of sound energy as it passes through it. Little energy is absorbed as sound passes directly through the wall but sound that enters through cracks on one side must bounce around in the insulation until it exits through a crack on the other side and that helps a bit but not anywhere as well as an additional layer of drywall or any kind of resilient treatment of the drywall.

Before you add the insulation look for sound transfer paths like back to back electrical boxes in the same stud bay and spaces over the top of the wall in the floor/ceiling cavity. Blocking and sealant are needed there; don't rely on insulation.

Because the cracks at doors are so large there is no reason to add insulation in the adjacent walls but it can't hurt.

X2.

unfaced friction fit bats work well..but better when sheetrock is up on one side of the wall.
faced batts will work too and give you a way
to staple batts in place.

remember to cut batts to fit around elec boxes,
and split batts...not compress around wires & pipes.
take your time & fit it well with no gaps/voids
or compressed areas.

best of luck.

Thank you for the responses. Sounds like quite a bit of work for first time DIY. It sounds like you both think it might not be worth it? I hadn't really thought about it being a problem before but my mother is pushing us to do it as she is wishing she'd done it in her house. Master bathroom is private enough but the guest bathroom is in between two bedrooms.
Thanks!

I think it 100% worth it.
just remember likd the other poster said that
things like elec rec boxes transfer sound also.

look at you house plan. any rooms that share
walls..bedrooms with baths/family room/kitchen
you want to isolate bathroom sounds & tv etc
from bedrooms. and nothing is more embarassing
than taking a leak with half a dozen people hearing.

just do a good job/ it isn't difficult.

best of luck.

Just remember that sound insulation is the least effective sound treatment. Pay attention to the more effective treatments first. If sound isolation at a wall is important to you, put a double layer of GWB on one side and seal the GWB at the perimeter and block and seal in the cavity above the wall.

Thermal insulation slows the horizontal movement of heat energy within a stud bay so it is important to fill every little space but sound insulation slows the vertical movement of sound energy within a stud bay so it is more important to fill the full width of the cavity to force sound to pass through as it bounces back and forth in the stud cavity. Think how effective insulation would be if heat bounced.

It is important to not think of insulation as a horizontal sound barrier as if it were a material with a higher mass like GWB or CMU. Insulation doesn't stop sound; it just wears it out which improves the sound rating of the wall a little bit.

One additional note about double layers of drywall. If you possibly can, uise different thicknesses. In spite of being mashed together tightly, they have different resonant frequencies and that will help on the low end.

We insulated all walls and lower level ceilings with regular batt insulation. The noise transfer from the upper floor was significant. We had water damage and had to replace the walls and ceilings in a lower level bedroom. After much research on sound insulation, we chose to use the Roxul, foam around all openings, and put "putty" around the electrical boxes. Plus used reslilient channel on the ceiling for the drywall. In my opinion there is a definite difference. The true test was a week ago when our son and dot-in-law visited with their little ones. They both said the room was much quieter. It cost more, but certainly worth it for us.