Are they testing for side effects in all three stages? I thought the first two trials were for safety and the third was for efficacy. Is that incorrect?

They're absolutely monitoring side effects during phase 3. some safety effects can be rare enough that they don't show up in phase 1/2.

Okay. I’m still having a hard time understanding how placebo helps them test for side effects.

I imagine: If a given side-effect occurs at the same rate in both groups, it's unlikely to be caused by the vaccine and is more likely to be psychosomatic.

Psychosomatic, or the result of administering an IV shot, or environmental.

Data on adverse reactions is captured by the CDC/FDA/HHS throughout the lifecycle of the vaccine, from all the phases of clinical trials to its use in the general population. Its required by law.