Background: COVID-19 vaccine side effects have a fundamental role in public confidence in the vaccine and its uptake process. Thus far, the evidence on vaccine safety has exclusively been obtained from the manufacturer-sponsored studies; therefore, this study was designed to provide independent evidence on Pfizer–BioNTech COVID-19 vaccine side effects. Methods: A cross-sectional survey-based study was carried out between January and February 2021 to collect data on the side effects following the COVID-19 vaccine among healthcare workers in the Czech Republic. The study used a validated questionnaire with twenty-eight multiple-choice items covering the participants’ demographic data, medical anamneses, COVID-19-related anamneses, general, oral, and skin-related side effects. Results: Injection site pain (89.8%), fatigue (62.2%), headache (45.6%), muscle pain (37.1%), and chills (33.9%) were the most commonly reported side effects. All the general side effects were more prevalent among the ≤43-year-old group, and their duration was mainly one day (45.1%) or three days (35.8%) following the vaccine. Antihistamines were the most common drugs associated with side effects, thus requiring further investigation. The people with two doses were generally associated with a higher frequency of side effects. Conclusions: The distribution of side effects among Czech healthcare workers was highly consistent with the manufacturer’s data, especially in terms of their association with the younger age group and the second dose. The overall prevalence of some local and systemic side effects was higher than the manufacturer’s report. Further independent studies on vaccine safety are strongly required to strengthen public confidence in the vaccine.