I suspect you are getting closer to what the “cause” may have been when you couple what Roger said 28 April post (adding plyo on top of everything else) and April 29 (too high a volume or too great an increase in volume of high intensity work) and what you state in your post about skate boarding and pushing almost entirely with the right leg, at least as it relates to the imbalance and possible why one side is suffering more than the other.
The side that you pushed with could have built up endurance/strength over time without you even knowing and I would think the number of impacts big or small will have made that side a little more “bullet proof”.
Precisely why the shin splints are more predominant on one side than the other? Could be a lot of things, you may plant the foot differently, one shoe may need an, or more of, an orthotic than the other (or position or composition of that orthotic), strength imbalance, natural differences from one side to the other of humans etc.etc.
In a related matter, IMO if you still wear flat skate shoes, GET RID OF THEM! If you have any foot or shin complaints those style of shoes may look great and be in fashion but they offer minimal if any support, more likely to cause problems if you tend to over-pronate or are somewhat flat footed to begin with.
If you continue to have trouble or even if you just want preventative ideas there are at least a couple of other threads on this complaint.