Add a layer of gypsum board each side of wall and insulate in stud cavity. Caulk the gypsum board all the way around and all penetrations with acoustic caulk. Locate door to mechanical room as far from bedroom as possible. Buy a quiet furnace. Water softener may have to be programmed to regenerate during day.

One simple option is to frame the shared wall on top of a 2x6 bottom and top plate. Then double up the studs and put one set facing the bedroom and one facing the mechanical room, use 5/8" sheetrock on each side, and insulate the cavity with rock wool . This limits the direct sound transmission through the wall.

If you use modern high efficiency mechanicals (HVAC, water heater, etc.), many will be set up for an external combustion air intake. However, some of these also use separate fans for the exhaust so that it is cool enough to use PVC pipe, so when they start up you will have more than one fan running.

There are also a lot of sound dampening/isolating products for drywall, mostly for folks who want to build a media room. Several I've seen use isolation clips so the drywall is spaced away from the floor joists or framing.

Bruce

Ditto Seaborn and Bruce, plus add a second layer of drywall to both sides with 'Green Glue' between the layers.

https://www.soundisoco.com/products/soundproofing-walls/green-glue/?gclid=Cj0KCQiAw_H-BRD-ARIsALQE_2MKo-DaJLjJ7ngj2a0jLQuw1HYElZiH2_Io-DiVT67U4DMdXjkqeM4aAt31EALw_wcB

Always start w/ the source.

- Isolate the equipment from the structure; by hanging w/ isolation hangers or using iso pad underneath for stuff that must rest on the floor. Make sure that ducts are connected w/ isolation joints.

- Cover the floor w/ isolation pad and then do a layer of plywood on top of that.

- Rock or plywood walls should be hung w/ isolation channel. If that's too expensive then do a layer of rock + green glue + rock or plywood. There are also pre-made sheets of rock that can work well.

- Insulate the wall cavities.

- Iso-Channel on the other side of the walls or use double layers w/ green glue.

- You MAY need to do some treatment within the room, particularly corners (base traps).

- Make sure that the trusses underneath this are extremely solid (talk to your structural engineer) as even w/ good isolation a furnace blower can generate noise through a wonky/bouncy floor.

So in our case w/ a loft mechanical room we have very solid trusses + plywood T&G glued & screwed floor + 3/4" isolation mat + 3/4" T&G floor (careful to leave a 1/2" gap around edges so it doesn't touch anything but the iso mat). Then the furnace sits in a drain pan that sits on top of an isolation pad (and that is a different frequency than what was used under the floor). MUA & Ventilator are hung from the roof trusses w/ isolation hangers. Walls use iso-channel & clips.