All drugs, no matter how well-researched, have some risks and side effects. Vaccines are no different.
The decision to administer any drug or vaccine can be thought of as a risk-benefit decision. In other words, the decision balances the risk of getting an infection and allowing the disease to run its course with the benefit of preventing the infection.
For licensed vaccines, this risk-benefit assessment is clearly on the side that the benefits greatly exceed the risks, even allowing for possible side effects and allergies.
Licensed vaccines undergo rigorous testing in laboratories and in clinical trials before licensure is granted by the Food and Drug Administration (FDA).
Even after a vaccine is licensed, the Centers for Disease Control and Prevention (CDC) constantly monitors occurrences of adverse events through surveillance systems like the Vaccine Adverse Events Reporting System (VAERS) and Vaccine Safety Datalink (VSD).