I really don't know what space being 3d means. Do people in this thread think that no one knows space is 3d? That the federation would ignore this fact.... If that was the case you could just "fly under" literally anything, and the whole world of Star Trek would be quite different if that was what everyone was doing. Everyone travels within this 3d plane to any place they need to, planets are not all lined up on a single plane, and we know there is a barrier that keeps people inside of the galaxy, so I don't know if you can go out into the void in any capacity, but I'm going to guess not for one reason or another. So no, people are not flying out of the galaxy, over to where they want, and then back down/up to where they want to be.
The real answer is that we see these races all of the time. Their owned territories don't stop them from moving. Space is enormous. You could travel in a straight line and not hit a planet just choosing a random direction. You sure as shit could go on as far as you want and never see a ship.
If you think "this is federation space" stops either of these groups, you don't know much about either of these groups. They don't give a shit. On top of that think about how hilarious it would be to try and stop literally everyone that tried to pass through your random blob of "territory." I mean really, think about how large those splotches are, and how ridiculous it would be to say "No one else allowed!" Think about the diversity we've seen throughout shows within the Alpha quadrant, how little these people actually know about planets and situations within this splotch.
If you've seen Voyager, think about them being in "Borg space" for instance. This doesn't mean there are just Borg cubes guarding every square light year. They went completely unnoticed for most of that time, and in fact, whole planets lived within that area. Some were taken out, some had long histories of fending off the Borg, others even just knew of them from what I remember, and hadn't really faced them. But, at the same time, this was definitely a part of space that was very much "borg" and you were at risk being within this zone.
I'm done typing, but to clarify, you're thinking of this like you would states or countries, but it's more like small tribes within a giant forest.
On top of that, I would like to point out that some of those maps are made by random people who don't know everything. I'm absolutely certain that "federation space" doesn't encompass everything from the edge of the galaxy to the center. If someone really had to, they could go around any territory in ST. The galaxy is FAR too large to really own that kind of real estate. Imagine it taking 40 years to get to the other side of your town, at the fastest possibly speed you have.