What could cause chest tightness on the right side, but not so much the left side?

Depends. Could be your right chest wall muscles or rib cage ... Could be your right lung ... Could be your heart ...
Many possibilities. Muscle strain, gastroesophageal reflux, peptic ulcer, gastritis, non-ulcer dyspepsia, gall bladder disease, pleurisy - to name a few. Heart disease can present this way too although it is, admittedly, a less common way. I suggest you see your doc.