Different martial arts take into account our natural sidedness in different ways.
- Jodo, and some classical judo instructors teach everything on one side. In jodo this is because of the inherent assymetry of the sword grip and scabbard placement - much of the material just wouldn't make sense on the other side. In judo, this is justified by saying that learning things on both sides would take more than twice as long and you'll never be as good on your off-side as on your good side anyway.
- Some classes (taichi and some karate for instance) teach the forms one-sided and leave the mirror forms or the off-side as individual homework.
- Some instructors obsess about teaching everything both-sided to try to minimize that weak side. This is a pretty good idea, but it takes a tremendous number of reps on the off-side to bring it up to par (if you ever can). It can easily take 3x longer to teach something both-sided as to teach it on the domnant side only.
Tomiki aikido has an interesting answer for this sidedness issue - kata that alternate sides between techniques. In some of the advanced kata (including ichikata and yonkata), you do the first technique right-sided, the second technique left-sided, the third on the right, and so on. This is a pretty good nod to the advantages of all the ideas above. You get some activity going on both sides, but you don't have to obsess about reducing sidedness to the point that you use up all your practice time bringing the off-side up to par.
Coming form arts and instructors that mostly fit the third style above - try to become ambidextrous - I have found the switching-sides katas irritating and frustrating for some years, but this training method is starting to make more sense to me these days and I'm actually beginning to enjoy the switching-sides kata.