Gun rights groups, in turn, are mounting initiatives of their own. In Washington State, where gun control supporters have already collected 225,000 of the more than 244,000 signatures needed for a ballot initiative on requiring background checks for private gun sales, their opponents have begun their own, dueling initiative, which would prevent the state from requiring background checks that are more restrictive than federal standards.

Gun rights supporters in other states are also focusing on legislators. In Nevada, for instance, a recall campaign has already begun against State Senator Justin Jones, a Democrat who led the successful passage in June of a bill to strengthen background checks. Gov. Brian Sandoval, a Republican, vetoed the bill…

But even some staunch supporters of gun rights concede that advocates of stronger legislation have won some striking victories.

“I think the Bloomberg side made very significant progress in some states,” said Dave Kopel, a lawyer and the Second Amendment project director at the Independence Institute, a research group.

“When you look at the state scoreboard,” Mr. Kopel said, “they’re roughly in a position of parity this year, which is doing well for them.”