Vaccine's are a mix of different substances. Some of those are in the vaccine because they are helpful for achieving the sought after result, other's are there because they are a by-product of the production process.
Side-effects from past vaccines often were due to preservatives and adjuvants. Given that mRNA vaccines get stored at -70 C they don't need any preservatives. Adjuvants are also not needed for mRNA vaccines.
Old-fashioned vaccines are often bread in eggs. That means that the egg protein ovalbumin is included in the vaccine which can result in a patient developing an allergy against eggs. It's worth noting that most of the problematic food allergies in our society are not ovalbumin allergies or allergies against other food proteins that can be found in vaccines, so while this effect is undesireable it's also no reason to reject standard vaccines.
mRNA vaccines get produced by hela cells which are a cell line based on human cells and as a result there's less reason to expect food allergy development due to mRNA vaccines.
It's possible that the COVID spike protein has some negative side effects but if that is the case, not giving COVID-19 becomes more important which makes the vaccine more important.
The nanolipid coating that surrounds the mRNA and gets it into cells could have some interaction that leads to side-effects but as far as substances go it's fortunately not something that interacts much with biological processes.
The only real serious worry I've heard about is antibody-dependent-enhancement, basically that in a worst case scenario a vaccine could make the disease more dangerous.
https://en.wikipedia.org/wiki/Antibody-dependent_enhancement
Just came across this clip from an interview with Paul Offitt that is relevant here: He claims that, out of all the serious side effects resulting from vaccines in the past that he could think of, all emerged within 6 weeks, so the fact that vaccine trials are required to look 2 months after the second dose before applying for an EUA should mitigate most safety concerns.
As another commenter suggested, one exception could be antibody-dependent enhancement (ADE), in which antibodies induced by the vaccine could enhance the severity of subsequent infection—indeed, this concern was not widely appreciated with the dengue vaccine Dengvaxia until years of post-licensure safety follow-up. But at least the specific mechanism of ADE that is operative in dengue is unlikely to be relevant with COVID-19 (ADE with dengue involves vaccine-induced antibodies against one serotype not being able to neutralize another serotype; the antibodies will still bind to the virus, though, and bring the virus to immune cells, which the virus then infects. But SARS-CoV-2 doesn't have multiple serotypes, and does not seem to be able to infect immune cells.)
I was chatting with friends about the Covid vaccine and they were concerned about side effects. I don’t expect significant side effects but don’t really have a principled analysis as to why not.
What kind of probability should one assign to significant side effects from a vaccine, particularly long term side effects which are less likely to be picked up in shorter testing regimes?