Kris's comment prompted me to look at the data on side effects. The Guardian has a great article on it, and there are graphs there that I'm not going to try to figure out how to reproduce that help visualize what I'm going to present textually below. I'm not sure how to use the table feature so my formatting won't be very nice. I'm using the Pfizer 2nd shot data because that's what we got.
Data come from
https://www.theguardian.com/world/2021/mar/18/covid-vaccine-side-effects-pfizer-moderna-johnson-is-it-safeSide Effect------Placebo group------Vaccine group-----%increase
Pain (inj. site)------12%-------------------78%---------------550%
Fatigue--------------23%-------------------52%---------------126%
Headache-----------24%-------------------52%---------------117%
Muscle Pain---------8%--------------------37%---------------363%
Chills-----------------4%---------------------35%--------------775%
Joint Pain-----------5%---------------------22%---------------340%
Fever-----------------1%--------------------16%---------------1500%
Diarrhea-------------8%---------------------10%--------------25%
Swelling-------------0.2%-------------------6.3%-------------3050%
Skin Redness--------1%--------------------6%---------------500%
Vomiting-------------1%----------------------2%--------------100%
I typed this all in manually, so there might be errors - check the link above if you want to verify. Keep in mind we're dealing with very small sample sizes, so we can't make very definitive assessments of the strength of the placebo effect here, but it is clear to me that the vaccine likely does cause side effects but that the placebo effect can be significant.