I am left handed and have been playing right handed for 10 years. I can now play along fairly well in sessions but I still have a lot of problems with bowing patterns, shuffles etc. I certainly think that the bow hand needs far more control than the fingering hand - if this were not so, then as Sibelius says above, right handers would have been playing the other way round for the past few hundred years!
It's still not a straightforward choice for a beginner though. It's very difficult to get a decent left-handed fiddle and you are restricting your choice of instruments in the future. I got a left handed fiddle after I'd been playing about three years but by that time I had got so far with the right handed one, and was enjoying the session playing so much, that eventually I just abandoned it and kept on with the right hand. I play well enough to enjoy the music, so I'm happy. However, I have since started playing both flute and concertina, neither of which causes a problem with the left hand, and I do find that I enjoy these more.