Side effects of vaccines are studied at many stages of their development and their use.
All new vaccine is tested in clinical studies, first with a few tens of adult volunteers in good health, and then with a few hundred volunteers, and then with several thousands or tens of thousands of children or adults, according to which vaccines are intended.
People or parents who agree to participate in these studies are everything which occurs from unusual in the weeks and months that follow vaccination, and the transmit their attending physician. They check these sheets and complement with their medical diagnoses.
Then all of these fact sheets are gathered and analyzed to compare the health problems that have occurred in persons who have received the new vaccine and those who have not received it. This allows to identify the common side effects more - those that occur in more than one person vaccinated thousand - before the vaccine been recorded and placed on the market.
Very rare side effects that occur eventually in a ten thousand, hundred thousand person or a million, may not be identified until this new vaccine is given to several tens or hundreds of thousands of people.
To detect these rare side effects, there are vaccines surveillance systems after their placing on the market. These systems work more or less well, because it is difficult to obtain all the statements of potential adverse effects. These spontaneous declarations are therefore mainly used to give a "signal", a suspicion that there may be a problem that must be considered.
These suspicions of side effects are then studied in large clinical studies involving tens of thousands of vaccinated subjects, and the same number of subjects controls, to determine if a medical problem occurs more frequently - or not - among people vaccinated than in those who are not vaccinated.
Indeed, vaccines do not protect against all health problems: it is therefore inevitable that these problems (diseases, accidents) also occur in people who have been vaccinated, just before or just after a vaccine!
The important question is therefore whether whether vaccination increases the risk of having a health problem - or not.