Dozens of police officers and medics have been given naloxone, a drug that reverses opioid overdoses, to treat suspected on-the-job exposures. Those affected have often complained of dizziness, nausea, and lightheadedness. Some pass out.

In Ohio, after a prisoner overdosed, 23 correction officers and four nurses were treated for possible opioid exposure, 31 inmates were relocated, and a nearby school was placed on modified lockdown.

“I never want to be in a position where I have to go see a family member, a wife, kids, and explain to them why their father or husband is not coming home that evening, or ever, for that matter,” Ryan Heath, the police chief in Alton, N.H., where Mr. Weil was charged, said in an interview.

“Everybody knows it’s a dangerous substance,” Chief Heath added. “I’ve seen it on the news.”

More recently, the Justice Department released another video, urging emergency medical workers to use precautions like gloves, eye protection and a face mask. In several cases, officers and emergency workers have conducted drug busts while wearing biohazard containment suits.

But such messages worry doctors, who contend that these precautions are unnecessary and could even do harm, stigmatizing patients — as happened in the early years of the AIDS crisis — and causing deadly delays while responders don protective gear.

“I want to tell first responders, Look, you’re safe,” said Dr. Jeremy S. Faust, an emergency physician at Brigham and Women’s Faulkner Hospital in Boston. “You can touch these people. You can interact with them. You can go on and do the heroic lifesaving work that you do for anyone else.”

True, a dose of powdered fentanyl the size of a few grains of sand can prove lethal, and carfentanil is even more potent. But the drugs must generally be deliberately ingested, not accidentally touched or inhaled, to cause a reaction.