After 18 months of fiddling with uncomfortable right-sided breathing, I went back to basics for a couple of months and today I was definitely very nearly just as comfortable breathing to the right as breathing to the left.
What's been good about rebuilding right sided breathing from scratch is that it's really helped me tune in to my clients who have various issues breathing. This was harder to do when I breathed mainly to my natural side as I did not have to think about it.
So ... my notes to myself for good breathing on the less natural side:
Head relaxed before, during an after the breath.
Focus on letting the water support the head when breathing and the head is on its side.
Arm depth: not too deep, not too shallow. Too deep pulls head down. Too shallow makes hips drop.
Arms parallel, don't let left arm be pulled across.
Don't think too hard or you end up breath holding.
I am a certified level 3.0 Total Immersion Swim Coach.