I wrote a little bit yesterday about sidedness in judo. In that particular system, the left side is mostly un-needed because there is sufficient variety of throws that they cover the openings left by other throws.
The way aikido answers this question is a bit of the same and a bit different. That is, we practice some things two-sided and other things one-sided. Tegatana is structured so that you encounter all possible combinations of motions. For instance, in sidestepping, we learn to step left twice, step right twice, step right then left, and step left then right. That type of practice helps to cover a lot of our sidedness options. Hanasu is simple enough to do on both sides, though it can be somewhat of a stretch to the mind of a beginner. Junana/nijusan is somewhat transitional in that you can practice both sides or not. Traditionally it is done one-sided, but some clubs do the off-side too. Owaza and the Koryunokata are longer, more difficult, and complicated motion-wise, so they are done one-sided. It turns out that if you do the fundamentals (i.e. hanasu) enough, your mind develops the ability to flip a technique to the other side on the fly.
If you want an interesting challenge to the mind, try doing nijusan with a one-armed fellow as tori. There are situations that you cannot get into unless uke attacks with his off side. So, for instance, in the atemiwaza, Patrick M. has to demonstrate the first two with uke attacking right sided, the next two with uke attacking left-sided, and the fifth one right-sided again. In the floating throws section, Patrick cannot do the sumiotoshi with the free hand in uke's face, so he cannot get into the three or four variants of sumiotoshi that come as a result of uke responding to that hand in the face. Patrick ends up doing a technically interesting mix of junana and nijusan.