I used to do everything right handed, unless I was working with a lefty, then we would mirror each otherr. Then this Summer I watched Bram Frank's excellent CSSDSC Knife series where he explained something that had become painfully apparent when I used to box.
We live in a right handed world. Left handers, in most circumstances, must train themselves to adapt to this world, manipulating everything from doorknobs, to manual transmissions, firearms, hand tools, etc. that were designed for right handed people. So now, instead of switching to lefty when I am training a lefty, I figure out how to adapt the drill/combat scenario/ etc. to where the lefty learns to deal with the righty - and the righty learns to fight the lefty. "Mirror Mirror" of lefty and lefty isn't that realistic of a training paradigm. Sure, it could happen, but if one wants to play the odds (and we all do, whether it is a friendly bet on a ball game, the craps tables, profiling, taking a chance on an illegal parking space, etc), then it doesn't make that much sense for left hander to train to fight another lefty.
To throw out some simple math, which everyone should take with a grain of salt, look at the following (assuming 10% of the population is left handed):
(liklihood of matchup)
Right vs. Left..... 9% chance
Right vs. Right..81% chance
Left vs. Right......9% chance
Left vs. Left....... 1% chance
And of course, if you are right handed, 90% of your opponents will probably be right handed. But if you are left handed, then 90% of your opponents will be right handed.
I had to take "Probability with Naval Applications" twice - and I barely passed it the second time, but I think these numbers tell us something about where our training time should be spent. It just isn't worth it considering how little time most folks have to train to teach lefties how to fight other lefties.
I am not advocating ignoring training the left hand at all - everyone should strive for a degree of ambidexterity - it improves coordination, helps keep the body balanced, and greatly increases survivability should the strong hand get injured. Hurt your strong side? Then work your weak side. Need a challenge? Switch hands for a while. One of my training partners used to play raquetball with his right (weak) side so his opponents would last long enough for him to get a workout when playing. I have a left handed holster for my 1911, waiting for the inevitable day when my trigger finger/hand/wrist/arm/shoulder on my strong side get injured... no reason to stop shooting just because I hurt myself in Judo or whatnot.
I yield the soapbox to the next person...
Dan