I've never heard that one side is the "correct" side to tighten the girth from. I've certainly seen plenty of people leave the girth attached on one side when they take the saddle off, and only tighten the other side (usually the left) when they put it back on.
I'm a little picky with how I girth up my horse since he came to me very girthy and it took us a long time to work through that. I never leave my girth attached to my saddle- Once the saddle is on his back, I attach it loosely on both sides and then tighten it one hole at a time, alternating sides. I also only use girths that have elastic on both sides. I've never liked the ones with elastic only on one side- it doesn't seem like it's really possible to get even pressure across the whole thing with that setup
I have seen plenty of long English girths that have elastic on one side only, but I don't think I've ever seen a dressage girth like that (maybe that's why the dressage trainer never said anything?)