The current match is at Carnaby Fudge, and the first stage is 'no sights, point and shoot'. Which I rather like.
Current handgun training tends to be 'always use the sights unless you're at contact range', and without question it's generally easier to hit accurately, even at close range, with sights. But point-shooting can be very fast and- with practice- very accurate. And considering the number of reports I've read where someone said they never really used the sights(in normal manner), I think skipping it is a bad idea. At least some familiarizing would be a good idea.*
James at Hell in a Handbasket has a good piece on it here, and if you look around the gunboards you can find more. If you have time, one book I'd strongly recommend is No Second Place Winner, by Bill Jordan. If you've been around the shooting world a few years, you've probably heard of him; he was a Big Name for a long time, with a lot of years on the Border Patrol and having(as one guy put it) ended more gunfights than most would-be tough guys ever thought about starting. The book's good for the stories if nothing else, but his chapter on 'Combat Style Shooting' is what we're after here. To shorten it quite a bit, the closer the bad guy, the closer you have the pistol to you when you fire. To illustrate his method, I scanned these two pages from the book:
If you've never tried it, or never seen someone who's in practice shooting this way, the accuracy you can get is amazing. Jordan would, for demos, stand aspirin tablets on edge on a table about ten or fifteen feet away, draw and fire(S&W Model 19) and shoot the tablets off without touching the table. The man who handled firearms training for OK Highway Patrol for years, a gentleman named Dan Combs, was about that good, and I got to watch him a couple of times. Flat damned amazing, both accuracy and speed.
I'll throw in, the way dad was taught(and he taught me) was to get both hands on the piece as soon as possible: so for anything beyond contact range to, say, five or six feet, use both hands as the additional control and accuracy is worth it. Jordan was a big, strong man(look at the hand holding that Model 19) and shot with one hand a lot, and could handle .357 Mag loads one-handed with no problem even in rapid fire; most of us ain't quite to that point.
That kind of speed and accuracy takes a LOT of practice for all but a few gifted people, but for the practical need of placing shots in an attacker's vitals at close range, just about anybody can pick it up with a bit of practice. It's a skill well worth working on. Let me close with a couple of paragraphs from the end of the chapter:
"One last suggestion: For 90% of your practice, draw from the holster and fire one shot. It's that first shot that is important and it is the one most difficult to place accurately. Don't practice "hosing" your shots, depending on seeing hits to get you on target. You learn nothing from this and you are lost if you can't see the strike of your bullets. Your crutch won't work at night or with no background to mark your shots, and then you will be in bad trouble. The first shot is all important, and if it is in, the others will follow.
For the other 10%, if you are concentrating on that first shot and it goes in you will have no difficulty with the rest of the burst. your wrist and forearm will stiffen automatically for recoil control. And above all, take al the time necessary but don't dawdle. Remember, "speed's fine, but accuracy's final"- if you are given time to display it!"
*A while back I read a piece by a police officer whose department had gone back to training on point-shooting after numerous reports of close range and/or low-light fights found that officers either didn't use the sights(seldom) or kept moving their eyes from sights to target(very often); if I remember right, the training improved their record quite a bit.
UPDATED It's been two years since I wrote this, and I've got a new post up here that touches on the subject. I'll also add that I think you should practice firing quick pairs in point-shooting, same as sighted fire; What I should have written here originally was 'fire single shots to learn point-shooting, and when you have the idea start firing pairs'.
And please remember: I'm not an Expert, I'm a guy stating opinions and thoughts. Don't think I'm the last word on a subject(I may luck out at some point says ego, but ignore that); if you want to try something I've mentioned, check out everything you can on the subject. There are very good books, like Massad Ayoob's and Jordan's, and nowadays lots of stuff on blogs(remember, some of those folks are just like me, while some of them really are experts), so there's lots out there to read on just about any subject or method you can think of. Do your research.