What happened is, one night I had some dry ice on hand for other purposes, and I thought about how the BME FAQs didn't have much information on using dry ice for scarification. Well, I was kind of curious myself, and I thought I could spare a patch of skin for the sake of science - so I decided I'd just find out for myself!
What I did was to pick a spot on the inside of my left calf, a spot selected less for aesthetics than for the fact that it was easy to see and work on. The only skin prep was shaving what sparse hair I do have around there. I picked a piece of dry ice about 1/2" square, and held it against my skin using forceps.
The sensation was pretty mild - sort of a stinging sensation, not very painful. I held the dry ice in place for about 30 seconds, by which time the skin had become yellow and quite hard. With no way to really tell, I'm guessing that the depth of the frozen area extended inward by about half the diameter of the surface area involved, probably in a roughly conical shape.
I allowed the area to thaw on its own, by the heat of the surrounding tissue. This took about five minutes, and wasn't painful. Once thawed, I could feel a vague stinging/burning sensation in the area, even though the frozen/thawed spot of skin itself was completely numb - damaged the nerve endings, I guess.
I had heard somewhere that in frostbite injuries, almost all injuries would recover in time unless they somehow got thawed out and then re-frozen. Since plain old skin isn't a very good scarification medium, this called for another step in the experiment. Using the somewhat reduced chunk of dry ice, I reapplied it to the same patch of skin, this time for about a minute. This time there was no pain or sensation of any kind.
Once it thawed out again, the area was reddened and had welted up a bit. At this point there was no pain, but it started to itch. At this point I covered the area with a bandage just to protect it from incidental abrasion.
Three hours later, the welting had gone down, but the surrounding area had started to swell a little. The next day, there were a few little blisters scattered around, which over the next few days sort of coalesced into one large blister covering the entire frozen area. This is normal behavior for a healing, uninfected frostbite injury - so far, so good.
After four or five days, I noticed that the center of the frozen spot, beneath the blister membrane, had actually become sunken-in relative to the surrounding skin.
After about two weeks, I had been poking at the blister enough to have broken it, and the injured area became somewhat tender to the touch - yep, infection! I finished removing the broken blister membrane, just to prevent the infection being trapped. The spot spend the next couple of weeks being oozy and sore, but eventually it started to clear on its own. Wound care consisted mainly of washing with antibacterial soap once a day, plus changing the bandage whenever it started to get goopy.
Finally, about a month and a half after the initial experiment, the spot formed a scab; after that, healing was completely uneventful other than the fact that it took much, much longer than I had anticipated. As I type this it is almost four months after I began. The scab is gone, the spot is shiny/smooth scar tissue, still red/pink, and still quite numb.
I started with a spot that was a 1/2" square, and the scar I have now is roughly a 3/8" circle. I don't know how this relates to scars acquired by heat-branding, but if you're thinking about getting branded, I wouldn't recommend freeze-branding for any kind of detailed work. Same as with heat-branding, stick to simple, bold designs that won't be ruined if it blurs a little.