https://www.scientificamerican.com/article/covid-vaccines-are-safe-and-effective-what-the-research-says/

As people around the world receive COVID-19 vaccines, reports of temporary side effects such as headaches and fevers are rolling in. Much of this was expected—clinical-trial data for the vaccines authorized so far suggested as much. But now that millions of people are vaccinated, compared with the thousands enrolled in early studies, reports of some rare, allergic reactions are surfacing, and questions are arising about whether any deaths are linked to the shots.

There is no question that the current vaccines are effective and safe. The risk of severe reaction to a COVID-19 jab, say researchers, is outweighed by the protection it offers against the deadly coronavirus. Nature looks at what scientists are learning about the frequency and nature of side effects as huge numbers of people report their reactions to physicians and through safety-monitoring systems, such as smartphone apps.