There are some good reasons to separate broodies. Other hens will lay in the nest with the broody, so you have to remove the new eggs every day. This stresses the broody, which is not good. She's under enough stress already.
The other hens getting in and out could break an egg. You really don't want a broken egg in a broody nest. It does not happen every time, but it can, especially if space is a little tight.
Some people have reported a broody returning to the wrong nest if another hen is on her nest laying an egg when she returns from her constitutional. She won't get confused if she has no other options.
Some people have reported other hens trying to lay eggs while the hatch is underway. A broody will give her chicks room to hatch without crushing them. A hen laying will not.
If the eggs are valuable to you, you might want the added protection.
I'm sure I'm missing a few reasons to separate them.
The main disadvantage to me of separating the broody is that she might cease to be broody when you move her. You always need to wait a couple of days after you move her to give her eggs to make sure she will stay broody.
Editted to add: You also have to have a place to separate them and set up extra feed and water. Make sure this separate area is predator proof.
With all that, in my opinion, you do not absolutely have to separate the broodies from the flock if you are willing to take those risks. I'm sure many others will strongly disagree. You set-up and circumstances come into play also. The tighter you are, the more reason to separate the broody, tight in both the coop and the nest.
People have been hatching eggs for centuries without separating broodies from the flock. Usually, these were people with a large enough flock that the loss of a few chicks in the hatch was not a disaster.
Editted to add: We never separated a broody when I was growing up on the farm. Ours were totally free-range. The coop door was never closed so space was never a problem. The chickens were livestock. Other than supplementing their feed in the winter with homegrown grain, they foraged for almost all of their own food winter and summer. We had a climate they could. If a predator showed up, it was dealt with by us as they are not very good at protecting themselves. Otherwise, they were expected to pretty much take care of themselves. We had enough chickens and eggs that, if a hatch went bad, we just let another broody raise a family to keep the chicken numbers where they needed to be.
I think it is a personal decision where you weigh the risks and decide what you want to do. I am building my coop so I can separate a broody, but that is my decision and I'll take responsibility for what happens. My set-up is different than on the farm.
