Doc has some great posts about this subject that have really helped to inform my approach. I still practice my techniques on both sides, but with what he's said in the past in mind I've realized that some movements are inherently more effective with one side of my body or the other. That isn't to say that I can't become more proficient over time with each side, only that some moves may have a higher functional ceiling with one side of the body or the other. At least, that's my understanding today, maybe over time I'll learn more.
-Rob
For me, I've only done it with a handful of techniques. One of my past Kenpo schools, had this as a requirement for the upper belts. Personally, I'd have rather seen other stuff being trained, but that wasn't my call to make. As for the Arnis...well, were I to have to deal with a left handed attacker, I'd opt for just keeping my stick in my right, rather than handing it off to my left, but thats just me.