As someone who used to believe in a round Earth having gone through the US education system (no, I did not grow up in a fundie compound), and then increasingly have doubts about what I was taught, I believe I can answer this.
To answer the main question, no, it does not mean everyone is 2D.
Also. To say the "Earth is flat" is probably an oversimplification.
So, we're going to use several objects to show you what I mean.
So, let's start with the globe. It's a three dimensional object, but I have a problem with it. I've explained why, but for brevity, we're just gonna move on.
Next we have a 2-D circle, of the sort you will find in math diagrams. As you can see, nothing inside the circle can stand up. This is a flat object, but any object near or within it is also flat. Moving on.
Next we have a table. A table is flat on top, but also circular, but you can sit on top of it (if you're crass and rude), crawl under it, or sit near it. It is a flat-topped, disc-shaped three-dimension object that has a straight edge for the objects occupying it. That is, objects can fall off the edge.
Next we have an old vinyl record (we could use a CD or DVD, but this works better). The record is 3-D as with the table but significantly more flat. Inserted onto a record player, the pin used to scratch it stands
upright, and rotates
around the circle (remember this, it's important). The record, then, can be said to be a flat-topped, disc-shaped three-dimension object that has a curved edge for the objects occupying it. That is to say, the pin used to scratch the disc travels around the disc on its top (I think it's clockwise towards the center, but I haven't really watched it). As it heading toward the center, and keeps scratching, yes it could eventually fall off, but if this object were traveling on a perfect curve and not essentially a spiral, it would just keep going in a straight line east (clockwise), never falling off or getting closer to the center.
Now, when we say "flat" we (probably) mean the model I drew above. This model of Earth has a domed atmosphere, and a dome which contains the rock, water, ice, and lava that would be beneath Earth's crust. This stuff under the crust also forms a dome, yet the crust itself is (relatively) flat. This means that like the pin on the record player, animals, plants, mountains, signposts, and rocket ships stand upright. This also means that like the record, all objects are moving around rather than directly through this crust. That is...
N = towards the center (until it reaches the center, then everywhere is south)
S = towards the edge (until it reaches the edge, then things get weird)
W = clockwise (sun rises in the east, and sets in the west)
E = counterclockwise
If you head east, you will not fall off the edge of the Earth. Same for west. If you start at the equator for instance, you will remain a 0 degrees N/S, never getting closer to the center nor to any edges.
The simplest way to visualize this is to view the globe from the top, which will almost get you Gleason's map. You can get from like LA to China by going north (how I actually did it on my trip to China) and from a top-down map, it is a straight line). Otherwise, you are going around the Earth to go west.