Whichever hand is [was] convenient is [was] the one the one that did the shifting, whichever shifting was needed at the time. Both with one hand, reaching through with one hand to shift the other side, whichever...

I'm right handed, and usually rear shifts are the more frequent. But not for rolling terrain, then it's both simultaneously for the most part.

But I no longer own any downtube shifting bikes. : (