A little back ground on the $0.02 I'll give you. I've been doing weapons instruction for the past 3 years, up untill a few months ago, I'm not an expert but I'm pretty dam good. Of this time, I ran a Marksmanship Training Unit for 13 months, doing USMC Annual Known Distance Instruction... I left that job and became the Cheif Weapons Instructor at a little unit, for another 13 months, instructing all organic Infantry Weapons.
For the Left handers, you can pick up shooting with your right quite easily, alot easier than most think, you've been battleing the right handed world all your life... it just takes a bit of practice and you'll be on your game. I've seen alot of left handers go amidextrous with great results, on both pistol and rifle.
Right handers... (thats myself included), it takes time and you'll be frustrated... but just keep going, for me I sometimes only shoot left handed at the range just to do it and not have the right handedness come into play, I've been putting more into left handed rifle than pistol lately, which has been helping my pistol game by leaps and bounds.
I'm trying to shoot rifle more from the left and not so much pistol left handed, I'm getting pretty good left handed pistol almost as good as my right, which has taken a few years, I still wouldn't shoot a match or steel challenge left handed yet, but that could be interesting. Left handed Rifle has been the biggest challenge, but it's something I've needed to improve upon.
Remember very few ever picked up a pistol/rifle and were naturally great, so switching to 'the other side', takes time and practice, that most aren't used too.
ETA: Shooting Off Hand Skeet and Trap... so far has been a total waste of clay, I've actually not taken shots just because I was so jacked up in the swing of the shotty, LOL!