Soon after their arrival in late December 2020, the COVID-19 vaccines turned the pandemic around and opened a path back to normalcy. They prevented about 14.4 million deaths worldwide, according to one estimate. In a small percentage of people, they also produced side effects. Over the course of more than a year, The New York Times talked to 30 people who said they had been harmed by COVID vaccines. Their symptoms may turn out to be unrelated to the shots. But they — along with more than a dozen