To answer your question straight out, it doesn't matter. If you are already playing the right handed way, just keep going with it. There aren't any left hand pianos, left hand keyboards, but that didn't stop lefties from playing them. The strength and dexterity will come with practice, as all guitarists must do to be good. I am right handed, and you might make the assumption that my picking hand would be stronger and more accurate. Here's my story..
When I had to relearn to play in April 2005, I began with alternate picking on diatonic scales, and my picking never really took off. By about Januray 2006, I had begun on Legato, the more serious technical 3 note per string stuff a la Joe Satriani. Within 2 months, my Legato had outstripped my my alternate picking which I had been practicing for 10 months. This goes to show, despite my left hand seemingly being weaker, I had discovered a technique that came more naturally to me than the others, something many guitarists experience. To some, sweep picking comes easy to them, to others, 8 finger tapping does, Legato, Alternate picking and so on. To this day, my sweep picking still sucks (well to some people I'm awesome, but 3 note per string patterns aren't that amazing IMO), not because of my left hand, which is pretty good at fretting the notes, but my right hand (my dominant hand) being out of sync with my left.
The trick is too find out what your weaknesses are, and work on them now. These days, I've been working on alternate picking so much, that I can play approximately 15.67 notes per second cleanly, and it's no longer far behind my legato.
Some questions: How long have you been playing?
What do you want to achieve with the guitar? (in terms of how far you want to get with it?)