The short answer is: you can't.
As an "honorary" member of a Burschenschaft, I fought two duels in Freiburg in 1988, leaving me with a prominent Schmiss on my right cheek (11 stitches) and a small one (one stitch) that probably wont be noticeable even if and when I go bald. I've probably witnessed over 100 duels. Though a lot of time has passed since I did this, I suspect things haven't changed very much.
Here's what you would need to do to even have the opportunity to get a duelling scar:
Join a duelling fraternity, or Schlagende Verbindung. Though I would later be offered full membership, I was able to become an "honorary" member of a Burschenschaft by virtue of a German great-great-grandmother (and because my father knew a German Navy officer who was an alumnus). A Burschenschaft is typically not going to admit you if you're not German. For the Burschenschaft that I belonged, you had to have completed your Wehrdienst, or compulsory military service - those who chose non-military service (at a hospital or rest home, etc) could not join. So you'd need to find a Landsmannschaft or some other duelling organization that would let you join. Most Verbindungen are NOT of the duelling type. You have to be a student at the university in order to join, but foreigners are more welcome in the Landsmannschaft and Corps organizations.
After joining, you would need to do a lot of training before you would be approved for a duel. Its all about correct form and style - the objective is actually NOT to slice open the other guy (nor is it to get hit yourself !!) - and maintaining it no matter what happens during the duel. Typically you'll have your "basic" duel ("Fuchs Partei") in which only basic strikes are allowed; the "Burschen Partei" is the second type of duel, in which nifty things like horizontal blows are allowed. Training for this kind of duel typically takes a year.
In a basic duel, often no blood is even drawn - and because the strikes must come only from above (in Freiburg - other universities can differ in their protocol), even if you are hit, the scar likely wont be visible afterwards. Its not until the advanced class of duels that things get interesting - the cheeks and foreheads (and top of the ears) are vulnerable to horizontal blows, which are much more difficult to defend against. Now, I'm sure someone, somewhere, at some point, actually WANTED to get hit during a duel. However, you're OBLIGATED to perform at your best, so willfully allowing your guard down in order to receive a sharp hit would be an egregious breach of protocol. But by the time you've prepared for an advanced duel, your training simply kicks in - there's no time to think about your reactions because of the extremely fast alternation of strikes.
As with every endeavor in life, people will have different skill levels. So opponents are carefully chosen by a council of representatives from different Verbindungen (you never duel someone from your own) and you will face someone the same height, speed, and strength. You don't get to choose your opponent, and you don't go around challenging others to duels any longer. Duels are carefully arranged and there is no antagonism between opponents.
Let me comment on a few other topics in the OP :
Duels are categorically NOT public events, and instead take place inside a Verbindungshaus, in a large room that can accommodate the duellists, their seconds, the guy who cleans off your sword after each round, the doctors, and many observers. Attendance will be restricted to only those from other local duelling organizations - recognizable by the unique color band you wear across your chest as well as your matching hat. If you're not from a Schlagende Verbindung, wearing your colors (and a jacket and tie), you won't be allowed in. Period.
There are no publicly announced duels.
Duels are not fought with sabres any longer (sabres have curved blades) but with straight blades of varying weights about 1m long. Only the tip and about a third of the blade is sharpened, on both sides. Its entirely possible to hit someone with the flat part of the blade, which actually hurts more than getting hit with the sharp part. A blade ("Klinge") is used for only one duel.
"I'm aware that I could go to a doctor or even do it myself….but this would not be authentic…"
You are correct. (Hmmm)
"Protective gear in modern times is nowhere near as baroque…"
Actually, Mark Twain's description is pretty spot-on - though most serious duelling groups will have made an investment in chainmail vest ($$$), which gives a lot more freedom of movement than conventional old leather. Also its easier to clean the blood off.
The "iron goggles that project an inch or more" nowadays have a nose protector that can be attached, but otherwise probably look the same as when Mr. Twain described them. A leather strap holds the goggles to your head, and its cinched very tight to prevent a blade from taking out an eye. The neck is protected by a heavy leather collar also cinched very tight. Upper body is protected by thick leather and sometimes chain mail over that. Striking arm is protected by thick leather padding.
The top of the head and the cheeks are left unprotected - these are the target zones.
There is no lunging around. Opponents stand one blade length from each other and only the striking arm is allowed to move. You cannot move your upper body or head at all. The worst thing you can do is to try and duck a blow - you're automatically disqualified if you do that. It's not good.
Getting hit doesn't hurt. Getting sewed up without anesthetic does. Forget painkillers since they would make you too slow. You aren't even supposed to drink 24 hours before the duel and if you get solidly hit, you'll only be allowed a couple of cold ones after the duel.
"most often and counter intuitively the left side…"
Actually, think about that for a moment; it makes perfect sense - a right-handed opponent will strike most of their horizontal blows to their opponent's left side. A wound on the right side is normally the result of two lefties duelling.
"how would a tourist go about arranging an authentic duelling scar…"
You just can't. And three months would be way too fast even for a member of a duelling Verbindung - in Freiburg people would practice for most of the year before their basic duel, and then would work a couple more months on horizontal strikes before being pronounced ready for an advanced duel (I gained some notoriety back then as the only American who had fought two duels as well as the fastest time between basic and advanced - 2 weeks)
And remember, not every duel results in a scar!
(Although I suppose you could pay someone to hit you in the face with a duelling blade)
Ironically, this Italian documentary film from the sixties is still the best I've found. Skip ahead to 2:30 for the duel portion.