Studies focusing on the safety and common side effects of vaccines play a crucial role in enhancing public acceptance of vaccination. Research is scarce regarding the usage of COVID-19 vaccines and the side effects experienced by health professions students in India and other countries. This study aimed to document self-reported side effects associated with COVID-19 vaccination among medical and dental students of six medical and dental colleges and teaching hospitals in four states (Tamil Nadu, Madhya Pradesh, Gujarat, and West Bengal) of India. A cross-sectional survey using purposive sampling of medical and dental students was conducted from 26 April to 26 May 2021. Data was collected using a Google Forms questionnaire capturing information regarding receiving COVID-19 vaccines, side effects and symptoms, onset and duration of symptoms, use of treatment to alleviate symptoms, awareness of haematologic risks associated with vaccination, and side effects from previous (non-COVID-19) vaccinations. The majority (94.5%) of participants received both doses of the Covishield/AstraZeneca COVID-19 vaccine. Among participants (n = 492), 45.3% (n = 223) reported one or more side effects. The most frequently reported side effects were soreness of the injected arm (80.3%), tiredness (78.5%), fever (71.3%), headache (64.1%), and hypersomnia (58.7%). The two most common severe symptoms were fever (14.8%) and headache (13%). Most side effects appeared on the day of vaccination: soreness of the injection site (57%), fever (43.1%), and tiredness (42.6%). Most reported symptoms persisted for one to three days–soreness of the injection site (53%), fever (47.1%), and headache (42.6%). Logistic regression showed that women were almost 85% less likely to report side effects. The study’s findings corroborate the safety of the Covishield/AstraZeneca vaccine’s first dose, evidenced by the relatively minor and transient nature of the side effects. However, the study underscores the necessity for ongoing research to assess the long-term impacts of COVID-19 vaccines, especially in the context of booster doses, thereby contributing to the global understanding of vaccine safety and efficacy.