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 experts — felt federal officials are not doing enough to investigate their complaints.

All vaccines carry some risk of side effects. More than 270 million Americans received about 677 million doses of the Covid vaccines, and even rare side effects — occurring, say, in just 0.001 percent of patients — might mean thousands of recipients were affected.

Indeed, more than 13,000 have submitted claims to a government fund that compensates people for Covid vaccine injuries. So far, however, only a dozen people have been compensated, nearly all of them for a heart problem caused by the vaccines.

Here are four takeaways from our investigation.

Even the best vaccines and drugs have some side effects. That does not negate their benefits, nor does it suggest that people should stop taking them.

The rotavirus vaccine, for example, is an unmitigated success, but it can lead to intussusception — a life-threatening condition in which the intestine folds in on itself — in about 0.02 percent of children who are vaccinated.

Some side effects caused by the Covid vaccines may be equally rare. Researchers in Hong Kong analyzed that country’s health records and found that about seven of every million doses of Pfizer-BioNTech vaccine triggered a bout of shingles serious enough to require hospitalization.

Other side effects are slightly more common. The Covid vaccines may lead to myocarditis, or inflammation of the heart, in one of every 10,000 adolescent males. (Myocarditis is one of the four serious side effects acknowledged by federal health officials.)

Deaths from the vaccines are vanishingly rare, despite claims from some conspiracy theorists that vaccines have led to a spike in mortality rates.

More intensive analysis may indicate that in some groups, like young men, the benefit of Covid shots may no longer outweigh the risks. But for the majority of Americans, the vaccines continue to be far safer than contracting Covid itself.

To detect problems with vaccines, federal agencies rely on multiple databases. The largest, the Vaccine Adverse Event Reporting System, is useful for generating hypotheses, but contains unverified accounts of harms. Other databases combine electronic health records and insurance claims.

These systems spotted blood-clotting problems associated with the Johnson & Johnson vaccine and a potential risk of stroke after mRNA immunizations, which is still under investigation. But federal researchers trailed Israeli scientists in picking up myocarditis as a problem among young men.

The American health care system is fragmented, with medical records stored by multiple companies that do not collaborate. Electronic health records do not all describe symptoms the same way, making comparisons difficult. Insurance claims databases may have no record of shots administered at mass vaccination sites.

Federal systems may also miss symptoms that defy easy description or diagnosis.

Among the hundreds of millions of Americans who were immunized against Covid, there were deaths, heart attacks, strokes, miscarriages and autoimmune illnesses. How to distinguish illnesses caused by the vaccine from those that would have happened anyway?

The rarer the condition, the harder it is to answer this question.

Merely judging by the timing — the appearance of a particular problem after vaccination — can be misleading. Most famously, childhood vaccines were mistakenly linked to autism because the first noticeable features often coincided with the immunization schedule.

Serious side effects may first turn up in animal studies of vaccines. But few such studies were possible given the nation’s desperate timeline in 2020. Clinical trials of the vaccines were intended to test their effectiveness, but they were far from big enough to detect side effects that may occur only in a few people per million doses.

Most independent studies of side effects have not been large enough to detect rare events, nor to exclude their possibility; others have looked only for a preset list of symptoms and might have missed the rare outliers.

An expert panel convened by the National Academies concluded in April that for most side effects, there was not enough data to accept or reject a link to Covid vaccination.

Federal health officials acknowledge four major side effects of Covid vaccines — not including the temporary injection site pain, fever and malaise that may accompany the shots.

But in federal databases, thousands of Americans have reported that Covid vaccines caused ringing in the ears, dizziness, brain fog, sharp fluctuations in blood pressure and heart rate, new or relapsed autoimmune conditions, hives, vision problems, kidney disorders, tingling, numbness and a loss of motor skills.

Some studies have examined reports of side effects and largely concluded that there was no link. Closer scrutiny may reveal that many, perhaps most, of the other reported side effects are unrelated to immunization. Most of them are also associated with Covid, and may be the result of undiagnosed infections. But without in-depth studies, it is impossible to be sure, experts said.