Some details about how the agreement will work in practice remain murky. The patchwork of state law means that states cannot necessarily restrict gun sales to everyone on another state’s no-gun list. And gun control advocates said states are already supposed to report to the national background check system people prohibited from owning guns under the parameters of state law, and not only those restricted by federal law.

But officials working on the coalition said the agreement would reinforce and expand what they share, including the names of people, for example, who have been voluntarily hospitalized for mental illness and are prohibited by some states from owning guns.

The agreement seemed poised to heighten the monitoring of people with mental illness, raising concerns among mental health advocates about unnecessarily stigmatizing people and discouraging them from seeking care. New York, for example, keeps a no-guns database that has grown to 77,447 names of people whom mental health professionals have reported as being a danger to themselves or others.

New York will now share that database with the other three states, Alphonso David, Mr. Cuomo’s chief counsel, said in an interview. It is not clear how the other states would act on it, given that the law in Connecticut and New Jersey, for example, expressly forbids firearm purchases only by people who have been committed to a mental health center.

But some states give their licensing authorities discretion to ban people they deem a risk to public safety, and Mr. David said they could use New York’s database as an investigative tool to make their own determination.