Why is it so hard to investigate the rare side effects of COVID vaccines?

Why is it so hard to investigate the rare side effects of COVID vaccines?

Nature, Published online: 01 April 2021; doi:10.1038/d41586-021-00880-9

For the vast majority of people, COVID-19 vaccines are safe and effective. But further research is needed to understand the causes of rare adverse events.Why is it so hard to investigate the rare side effects of COVID vaccines?
Source: Nature