Huh. What do you know. A Dyson swarm made out of a torus or two is going to have a slight atmosphere. Not much you can do about it, some gas is going to go astray. The surface of a torus is functional; it captures sunlight and radiates waste heat. And it holds in the atmosphere, either that or the atmosphere (and the torus as a whole) slowly loses mass. Outside the surface is a more dangerous zone, anything out there is liable to be zapped unless the inhabitants of the torus have given it clearance. Lasers can zap things a long long way off, provided there's no atmosphere.
So the surface of the torus is going to be a solid membrane, something like a mylar balloon acting as a solar cell and holding in a very slight atmosphere. Somewhere between 1 pascal and 1e-8 pascals. It has to stretch. One side of it also has the pressure of the solar wind and solar radiation, which is about 1e-5 pascals.
The surface can't be in net tension ... it could have some bracing that can adjust its dimensions as the shape of the surface changes over the course of each orbit. I can think of several ways to do that with stretchable fabric and jointed sticks, not even invoking nanotech materials. The inside of the torus will have a slight atmosphere. Everything on the inside won't have a clear view of the stars. Instead there will be a haze with distance, like our daytime blue sky. Unless the atmosphere is very slight indeed, seeing the whole interior of the torus ... which is like the distance from here to Saturn ... won't be possible.
Having an atmosphere doesn't sound desirable, so there will be efforts to clean it up, so it will be very slight. An easy way to clean it up is to have a planet in the core of the torus, where the mass of the torus and the temperature of the gas are such that the planet naturally accumulates atmosphere faster than it loses it. I imagine hydrogen and helium are the most common, and hardest to contain, so they'll be the majority of the atmosphere. The interior will be slightly lit, too, for the same reason it has an atmosphere, much like light pollution in today's cities.
Every point in the interior of a torus has a prescribed speed and direction that keeps it in an orbit with the same period as the rest of the torus, with neighbors always going the same speed and direction as neighbors. So a torus is immune to the Kessler effect. It's likely that neighbors will be tethered to neighbors. Still, in something this big with heat differentials, there is going to be some amount of weather, with the wind not going quite the direction you want it to some of the time.
The atmosphere says that something inside the torus going at very much the wrong speed is going to slow down to the right speed after awhile, all on its own, no tethers or calculations necessary. That also means that jumping from one point on the torus to distant other point isn't just a matter of choosing the right initial velocity and direction. Ditto for laser communication ... it decays with distance. Communicating along the surface with relays works, of course. Jumping from one point of the torus to another on the outside of the torus is likely to still work.
Some rough math on wolframalpha tells me a torus centered on earth's orbit with a surface .1mm thick (about the thickness of a mylar balloon) will take 5e19 cubic meters of material. Earth's moon contains 2e19 cubic meters of material. So, the surface is going to take a sizeable amount of matter, but still probably a pretty small fraction of the torus's total mass. The interior of the torus will be nearly as empty as space, but that still allows for zillions of things to be scattered about in it. I expect most stuff will want to be either near the surface or near the core orbit.
How big will Dyson swarms be? If you put one around the sun, and radiate at double the universe's temperature (2.7*2 = 5.4 degrees Kelvin), and follow the Stefan-Boltzmann law of power radiation, the swarm has to be 5700 AU in radius to radiate the sun's energy like a 5.4 degree Kelvin blackbody. Pluto is 32 AU away. The mass of a 0.1mm surface scales with the square of the radius, so a torus 5700 AU in radius would have a surface mass equal to the sun. That's too much. Either Dyson swarms will be hotter than that, or they're going to choose to be around much cooler stars.
If you put one centered in earth's orbit, 1AU in diameter and 1AU thick, it's about 4e23 square meters. If you put it 5AU from the sun and made it 4AU by 6AU thick, it'd be about 1e25 square meters. If you built the one 5AU out from mars and jupiter (handwave about how you can construct anything out of nearly pure hydrogen and helium), that's 2e27 kilograms, so it'd be 200 kilograms per square meter. That's a lot thicker than a mylar balloon, but it's still exceedingly flimsy on this scale.
Ever seen a super-big soap bubble? It doesn't stay spherical, it billows and splits due to slight gusts of wind. These toruses will have that problem due to the atmosphere and solar wind and passing planets, unless some form of station-keeping stabilizes it. The easiest station keeping techniques is to either capture sunlight or not, and either hold in the atmosphere or not. But making use of these would either not capture all sunlight, or release the atmosphere to space, or both. Throwing rocks might work. Electric currents and magnetic fields might work.
The further the torus is from the sun, the colder it is, and the easier it is to capture stray gas. But also the bigger and emptier the torus will be, and thinner the surface is required to be.
A Dyson swarm with two toruses at right angles would have a hot bright inner torus and a cold dark outer torus. The outer one is bigger, thinner, and receives less energy, so they'll have rather different economies and cultures. They're likely not to like one another very much.
You would surely arrange it so that the inner surface of the outer torus is travelling in about the same direction as the outer surface of the inner one, to minimize the damage from stray junk drifting from one to the other, and to minimize the energy needed to jump from one to the other.
Perhaps the inner torus would have eccentricity = tilt, and not be that thick a torus leaving a big hole, while the outer one would have tilt much more than eccentricity so it has a relatively small hole. A thinner inner torus allows more light to reach the outer torus, and the outer torus doesn't have as thick an atmosphere so it might worry about weather less so the extra tilt would bother it less. The contraints on a Dyson swarm made of toruses are just that the thickness of thickness of the inner one is enough to block light from escaping the hole of the outer one, and the thickness of the outer one is enough to block light escaping the hole of the inner one.