From a street fighter. I'm right-handed, 6'1, and 160 lbs. Like the others I have no scientific facts to support my claims other than experience. In real fights, you are typically (if not the aggressor) in the position of defense; if you survive you win. If you are on the offense, total submission is required to call it victory.
It has been my experience that in those situations both hands should be trained; though I hesitate to use that word, since to train unless strict attention is paid to actual scenarios, i.e. sparring, wrestling, and throws done with the view of true combat, they are robotic programs that will crash upon being struck.
No one lets you punch them out or throw them around or pummel them without a fight. When you practice keep this in view. Mix up your partners; short, tall, fat, skinny, agile, clumsy. I can't tell you the number of times I've gotten hit by the clumsy guy with no technique rather than the well trained "black belt," who attacks by rote, with a form or kata, or some predetermined move.
That stuff will get you in real trouble if your in an actual life or death situation. Mostly I use south paw though right-handed. This is for the reach (for throws and take downs) but I mix it up and if I'm fighting a lefty I usually adopt the regular stance (left foot forward) to mirror my opponent. This takes some training though as others have mentioned, the main thing is not to be caught in an unfamiliar or compromising position.
The mistakes I typically encounter are:
Feints that are not followed up by attacks.
Don't waste time on stuff that does not affect your opponent. This also can be a disadvantage since to an experienced fighter the faint can show you a lot about your opponents style and lines of attack so be wary of using it.
Trying to land a favorite punch.
Watch the Mike Tyson vs. Evander Holyfield fight for details. So many times I've been in a fight with someone right-handed, 6 foot plus, over 200+ lbs. with loads of upper body strength with a preference for the right cross or right hook, who they'll try and turn every encounter into an opportunity to throw that favored punch. That's an almost guaranteed way to get your hat handed to you. Same goes for combos if you can't throw your right hook without first setting up with the left jab or something else, then you'd better hire a bodyguard. There is no one punch/one kick fits all solution. The best fighters, the ones you really don't want to fight in a dark alley are guys who have no formal training and therefore; they have no hang-ups nor preconceived ideas.
They get up when you knock them down, and are most often a greater threat than men skilled in one particular set of routines. I'm not knocking any particular martial art. I've sparred with most of them who practiced tai chi, kung fu, krav maga, akido, sebukan to name a few. The best of them, when sparring, almost threw their formal training out the window and hardly a "traditional" punch or kick was thrown.
You've seen that guy that gets red in the face lowers his head and charges, or that guy who throws wild haymakers without aim or intent. This is an easy victory to the man who has any street fighting under his belt or judo for that matter. In the first situation, let him come and his momentum will do the rest. You can apply total stoppage to that momentum, as in a knee to the lowered face, or let the momentum continue until they are committed and then side step and throw or take down, etc. If you are not good at your ground game, don't take guys to the ground. Just throw them down and get ready again. It wears down their ego as well as there body.
My final word of caution is don't get cocky. There is no defense against a well timed and properly aimed attack. Be patient but never hesitate. Try to react without thinking but out of instinct. By all means give ground; it is the fool who can't give a yard to take a mile. Unless your back is to the wall, moving backward is just another direction. Though like the feint simply retreating is not going to win anything. I find that if I retreat for the first couple encounters I can get a feel for their rhythm and disrupt it. It's like in music even the rests are counted, and when played properly every note in the song has meaning.
Every move your opponent makes reveals more about him. Learning to recognize and counter those movements is the key. For me in my training since my muscles are not for show but for work, I don't go in for fancy or elaborate workout routines. I run every day I can. I chop wood for practicing my aim, especially making kindling. I spar with as many different types of guys as I can. I eat what I call the homeric diet, cut out the bread from your burger and you're close. Avoid being stuck with only one option, this goes back to your question. There are countless situations that could arise wherein your strong hand is out of the picture. What then? I don't care how good you are if your only good with your right hand. You will betray yourself in real combat. Your stance and your moves will indicate to someone like me your favorite hand, and I will break it should the opportunity arise. What then? As a street fighter the only rule I honor is "there is no such thing as a fair fight."
If you have an advantage take it. If your opponent has a weakness, exploit it. If you only have one option, i.e. your favorite hand, what happens when I break it? You may discover hidden reserves of strength and talent latent in your hitherto unused limb but what do you think the chances are of that happening? Better to train both hands so you'll have something to fall back on. All of which is to say better to have no training than to be a slave to your training. The more you fight or lacking that, spar, the less you do and the more you be? You react less and respond more, you intuit the next move from the one that preceded it. you don't waste time or energy. and formalities disappear.
I am lucky that I have a twin brother who is left handed where as I am right, so I get a lot of experience fighting lefties. I am good at throws but don't really like the ground too much, where as my brother can fight on the ground like nobody that else I've seen, but he doesn't like the long game. Both of us train our weak arms as much as we train any other part, which is to say that I don't train, I practice, I fight and spar. For out and out beginners I suppose that training in the mirror will improve execution, but against what? When you kick the air it doesn't drop down and smack you in the balls or trip you. How can you be sure that that timing and force were correct?
Still I can count the times that I knocked I guy out with my favored hand on my hands, but I couldn't tell you the number of times my left hand and being proficient with it saved my ass.
So yeah train both hands bot not against wood or the air but against a live and reacting opponent. I recommend Bruce Lee's works for further reading especially if you are not at the beginners stage, and fencing and boxing for foot work.