All of those techniques will help and which is better depends on which manufacturer you believe but if sound travels through the existing wall that easily you obviously have actual physical leaks at cracks and penetrations even at the top and bottom of the wall or perhaps above the ceiling.
So pay attention to that aspect of sound transfer. Any wall is only as good as the seal at the edges and at penetrations. For instance, there is little you can do if there is a standard door in the wall.

Using properly designed sheathing on resilient clips will provide the best sound attenuation. Additional benefit will be achieved by using sound deadening insulation in the stud spaces. Finally, seal around all penetrations, ie, electrical outlets, switches, etc in the subject wall.

If you really want to deal with the sound transmission, replacing the gyp board using resilient clips will be the only really effective solution. Everything else will be a bandaid approach. Good luck!

Odd that the sound is traveling that easily. Agree with checking around the penetrations. Sound could be going through HVAC registers or maybe adjacent to electrical boxes (if near each other on both sides, etc.)

I found out that the HVAC registers (which are only a few feet apart on either side of the wall) for the 2 rooms, come out of a common metal "box" which one flex duct comes into. The AC guy told me this is contributing to the sound problem and he can cut the box in half so that each room will have its own flex duct/box and not be connected to each other that way. I am hoping this will help. I have researched the wall soundproofing, and I think the best bang for my buck will be using GreenGlue to add a panel of Type X on one side of the wall, which I plan to do if the HVAC fix isn't enough improvement. Thanks!

Type X drywall is, by definition, drywall that provides a 1-hour fire resistance rating when one layer is applied to each side of wall studs. As it happens, that turns out to be 5/8" drywall that has internal fibers that prevent the drywall from collapsing when the required hose stream test is conducted after a one-hour exposure to fire in the required test. Unfortunately, Type X has no greater sound resistance than any other kind of 5/8" drywall.

For better results use one layer of drywall on one side and two layers on the other whatever the thickness of the individual panels.

As I have already mentioned, sound transmission through walls is commonly misunderstood by homeowners and the internet is full of information from manufacturers that think only their product is effective. The fact is that most sound passes through walls through cracks, openings, and adjacent spaces above, below and to one side of what appears to be a solid wall therefore adding more drywall is only helpful if you have addressed these issues as well. An acoustic engineer will tell you that sound abatement techniques often do not show the expected results in field tests because the way the wall is built is so often more important than what it is made of.

Walls are never "paper thin"; if they seem to be that way it is always due to sound leaks.

Renovator8, after I install the additional layer of drywall, I will have one layer on one side and 2 layers on the other side: 1/2" on one side and one 1/2" + one 5/8" on the other. Are you saying that I could make the second layer 1/2" just like the others, with no difference than if I had used the Type X 5/8"? I am planning to use the acoustic seal around the perimeter and outlet boxes when I install the second layer.

What I was telling you is that "Type X" only means the drywall is "fire-resistant" but it has no better sound resistance than regular 5/8" thick drywall. The same is true for the more modern Type C fire rated drywall.

The thicker the drywall the better and using an imbalance of thicknesses on each side also helps but serious sound abatement usually involves mass (thickness of drywall), resilience (flexible clips/channels or green glue), absorption (insulation in the cavity), and sealing of cracks and penetrations.

Sound energy moves in a very efficient manner so it requires more than one detailing technique to stop it.

I was wondering if you tried the fix and had any reduction in sound. We also have thin walls, esp in our bedrooms. When standing in one, I hear most of the words the person in the next room are saying. Also, our family room shares a wall with the bathroom. When the TP is being pulled off the roll, you hear it. It's very disconcerting as this is a nicer home in a good neighborhood. I just want to know what worked.
Thanks

"Thin" walls are almost never the cause of excessive sound transfer in a home so it is misleading to use that term. A well designed sound resistant wall can be completely compromised is the electrician puts outlets back to back in the same stud bay or the HVAC contractor allows a return air duct to serve adjacent rooms in the same stud bay or if the ceiling or if the drywall on a shared wall stops at the ceiling. Sound will always find the weakest part of the separation.

Get rid of the leaks before you even consider adding drywall and making the wall "thick".

Unfortunately, there are no simple fixes for so-called "paper thin" walls.

And detailed research at the National Research Council of Canada shows that flanking sound transmission--over and below the wall and through the floors can sometimes never be significantly attenuated without major structural changes.

In other words, you can bulk up the wall all you want, seal all the cracks and the walls may still sound "paper thin."

See, for instance, the diagram below illustrating that even in walls detailed with resilient channel and multiple layers of drywall, the apparent STC is way lower than the wall transmission class. (Lower is noisier.)

Source: Guide for Sound Insulation in Wood Frame Construction--Part I: Controlling Flanking at the Wall-Floor Junction p. 28

If sound reduction is a priority with the homeowner, it has to be carefully planned prior to construction so that every step from design to completion is consistent with that goal.

For instance, floor plates should be caulked before being installed; same with drywall.

As mentioned by other posters, HVAC can short-circuit the best-laid plans. Especially if one is using "best practices" in providing air returns! When I've built for myself, I've deliberately sacrificed the most efficient HVAC for sound isolation.

Google wool slag insulation.