Do I need to soundproof this wall?

What I see is a wall that is firmly anchored to the floor and looks to be weight bearing at the top. It will transfer vibration both ways. How do you plan on finishing your walls and ceiling?

I like this one just put a step up to riser height outside the room in that hall and build the risers from side wall to side wall. That will allow you to center up the seating and create an aisle on both sides.

Invest an hour to read all the articles on soundproofingcompany.com. Pay attention to the topic of flanking pathways. It is not blocks you want it is whisper clips and hat channel. Double layers of 5/8 drywall with Green Glue in between.

IMHO this is the kind of walls and ceiling you need in a townhouse with neighbors. This is a video click picture

The fact that the studs do not touch the concrete is a great feature because they can't transfer sound except at the top and bottom. This is very common in residential basement finishing. By code untreated wood studs can not be in contact with concrete. So you are off to a good start, now you need to add Mass (Multiple layers of drywall), Dampening (Green Glue) , Absorption (insulation) and the final touch would be to mount the wall on springs for Mechanical Isolation (Whisper clips and Channel)

Adding an additional layer of damped drywall (Green Glue) is more effective and far cheaper than hanging a layer of MLV

Now when you build this room you don't want to cut a bunch of holes in the drywall "Bunker" for lights or electrical boxes. Figure out a design where everything can be mounted inside the room. Where impossible utilize backer boxes or putty pads.

Mass loaded vinyl:

1. The only thing MLV brings to the table is Mass. You'll get more mass for less money from drywall;
2. The vast majority of pictures showing how to install mass loaded vinyl, if followed, will increase sound transmission, not decrease it.
3. As Big said ... read about flanking paths. Doing just the wall will be somewhat helpful; but, in the end a big disappointment.

You are absolutely right, I thought that was concrete from first glance but it is actually drywall. The question is, is that the neighbors actual wall or a fire barrier extending from the foundation up to the roof.

Since it is not concrete it is even more important to pull out the stops on soundproofing.
If that is a fire barrier partition and not the actual wall there is another concern that once you put a wall up you will create a triple layer which has some negative ramifications. This is discussed on the web site I suggested.

You might call the local Building Inspector's office. They often are aware of common construction practices in different neighborhoods.

You said you were finishing out this room. To be a theater? Is that the kind of sound we're talking here?

EDIT: Just saw your thread.

I'd consider treating all 4 walls and ceiling.

I'd either apply clips and channels to that wall or build a new wall in front (preferable). Double 5/8" drywall or comparable mass everywhere.

Before you start framing anything, it would be good to determine your HVAC plan. Also the door plan. Both of these are big holes, and after all, life already has enough big holes.

yes I'm just down the road off 7 and Reston Avenue.

I think you might get friendly with one of the neighbors and ask them to go down to the basement and knock on their side of the wall to figure out what you are working with.

On the other three walls, you need to keep in mind that sound doesn't just travel in a straight line.

A room full of people and a space heater disguised as a projector will get hot even in the winter. In addition to the supply, you need to figure out a return for the room to keep the air circulating.