In judo, the motions in different throws interact with our sidedness in different ways. In other words, there are different kinds of sidedness in judo.
Universal Dominant Throws
There are a very few very versatile throws that are easily learnable with minimal training and are throwable with the dominant side whether the opponent is moving left or right, forward or backward. The only universal dominant-sided throw I have in my circle is osotogari. I can throw it right-sided whether the opponent moves forward, backward, left, or right. The opportunity for right-side osotogari literally happens on every step the opponent makes. That doesn’t mean that I can necessarily pull it off every time, but the opportunity is there.
Universal Variant Throws
There is a larger set of throws that can still be thrown on every step forward, backward, left, or right – but different variations are required in different situations. For instance, I can throw deashibarai on any step the opponent takes, but if I catch it early in the cycle it happens with one leg and if I catch it late it happens with the other leg. Two variants of deashi that are equally effective but together they make deashi possible everywhere. Hiza is similar for me. I throw a left and a right variation of hiza as well as an early and a late variation, which makes hiza pretty much a universal throw for me.
There are other throws in which the motion is so simple that they are truly ambidextrous. For example, the drag-and-drop kosotogari that is so effective for me in randori is so simple in its action that my natural sidedness never comes into play. It is trivial for me to do left-sided, so I also have no problem right-sided.
Special Situation Throws
There is a fourth class of throws that I can only do in special situations or on a particular side. These special-purpose throws seem to fill in tiny holes in the system where it may be easier to throw in this particular way than using one of the universal dominant or universal variant throws described above. Virtually all hip throws are this way for me – particularly the one-legged hip throws like haraigoshi.
Now, my premise in a previous post was that it is basically a waste of training time to obsess about overcoming sidedness. For instance, I listed haraigoshi as a special-purpose throw above. Obviously it is possible to learn to do it on both sides – you have to do both sides of it in Nagenokata. Junokata also gives good practice in left-sided hipthrows. But I still maintain that in training people to respond to general randori/shiai situations it is better to train the hipthrows predominantly on one side, whether it be arbitrarily chosen as right side or whether it be the player’s dominant side. In addition to taking immense amounts of training time to bring the off-side up to a satisfactory level of execution, it is just un-necessary because of the existence of the universal-dominant, universal-variant, and ambidextrous throws.
Of course, the usual disclaimer applies. This is how my judo seems to work for me right now. There are lots of people who could beat me up in shiai and could out-teach me too. So, your mileage may vary.