I don't know what a lot of people means. Hundreds? Thousands? Millions? And certainly there's been debate in the medical community about how many of these cases are caused by the vaccines or were there other factors that triggered a given condition.
They weren't trusted without question. There were multiple extensive clinical trials completed before the vaccines were rolled out. It's fair to ask questions and wonder whether the vaccines were rolled out too quickly, but there certainly was oversight and scrutiny to make sure the vaccines were well tested and as safe as possible. I trusted the process more than the companies themselves.
Pfizer has brought some very beneficial drugs to the market over the years and they've also made some mistakes and worse at times. That said, I trust my doctor more than I trust a given pharmaceutical company. Pfizer and Moderna had the resources and capacity to bring the vaccines to market and there aren't too many other companies out there capable of pulling that off in the timeframe they had.