I've noticed that with dumbbell bent over rows that I can do more reps even with higher weight on the right side than the left. The same is true with the dumbbell shoulder press. Should I use the same weight on both sides, waiting for the left to catch up before increasing weight, or should I increase each side independently as I'm able?
The former - let your weaker left side catch up to your stronger right side. You'll never get them to be exactly equal, but a wide discrepancy in strength between the two isn't good, especially if you plan to get into activities outside the gym.
This is common, BTW - if you are right handed, you use it much more than your left. I've made it a practice that when I'm carrying things (groceries, for example) I'll use my left hand to carry the heavier bags.