Which plant you're talking about definitely makes a difference. How careful you have to be with the roots depends on the species. Some are very sensitive and some are not.
With tomatoes, if your plants are healthy and are not tiny, you can just pull one of them up like a weed and repot it. (Tomatoes are very lenient with their roots.) Then water the old one and make sure soil is covering its roots. They should both live and be fine. They should develop new roots quickly. I've done this a fair amount. It works with tomatillos, too. It may stunt peppers for a while, but that can be overcome with the right nutrients and growing conditions. Potassium sulfate is an example helpful nutrient. Bright light (after the plant has adjusted a little, and not before) is an example helpful growing condition.
I'm not saying pulling it up like a weed and repotting it is the ideal and best way to do this, but it certainly works with tomatoes and tomatillos. If you don't have potassium sulfate and you want to be extra careful, you can put the one you separate out in water for a couple days. This will help the roots to grow out and recover a little, and the transplant shock shouldn't be as bad.
You might think this is a lot like taking a cutting, but your success rates will be higher here than with a cutting, even if there's damping off in the soil and there are hardly any roots on the plant. (This information comes from my experience. I'm not sure why this is true, though. I guess broken roots are different than a snipped stem without roots.) Nevertheless, I would avoid the damping off pathogens to be safe, if you can.
Other plants may not be so forgiving as tomatoes, tomatillos and peppers. Tomatoes, tomatillos and peppers all grow roots all along their stems (not just at the leaf nodes). They have a big advantage. So, replant them deeper, unless they're grafts. You don't want the portion above the rootstock under the soil or else it may defeat the purpose of using a disease resistant rootstock.
Now for somewhat trickier plants, I would guess (and for now, this is just an educated guess based on what I've seen with less tricky plants in various situations) that the following would help: after separating your plants, soak both of their roots in water for a couple days and repot them afterward. If they survive in water for two days (which they just might in water, more easily than they would survive in soil directly after being disturbed), they'll likely survive the transplant afterward. Give them potassium sulfate (maybe even in the water while soaking, too). It reduces transplant shock.
If you separate them and you don't soak them in water, make sure to keep them out of bright light for one or two days (and put them back in bright light afterward, if that kind of plant likes bright light). If you do soak them in water, they may not be as bothered by light, but watch for wilting just in case. When light makes a plant wilt, that's very bad. Take the plant out of the light ASAP.
I can't guarantee this would work for trickier plants, but I would try it if I could afford the risk. In fact, I'm going to. I've got some eggplant, ground cherry and litchi tomato plants that are in this situation. None of them seem to grow root nodules along the stem in humid conditions like tomatoes do, and I've heard some of them (litchi tomatoes and ground cherries) are difficult to transplant. After my soil amendments arrive (worm castings, mycorrhizae and more potassium sulfate), I can try it out. Rather than repotting after soaking, I want to plant them outside, since I'm low on space, soil, containers, and time. They'll probably be covered with milk jugs afterward (so hardening off shouldn't be a huge deal). I try to avoid hardening off where possible, but I'm not saying you should. Anyway, I should be able to add more to my answer in a week or two. Feel free to remind me in a comment if I forget.
For much trickier plants, or if you just want to know for sure that you'll save one plant, you might be better off snipping one off and treating it as a cutting. The other one should be fine, in this case. However, for the one you cut off, I would personally try to cut it off so low that you get at least a root or two. Then your odds of success with that 'cutting' will be much better.
Alternately, you could just leave both plants together forever.