ST. PAUL — A half dozen police chiefs and sheriffs argued Tuesday in a packed Capitol hearing room that Minnesota isn’t doing enough to protect against gun violence, kicking off three days of hearings on a host of new proposed limits on firearm ownership.

Hundreds of people from both sides of the debate swarmed the Capitol office building for the hearing, jamming the committee room and several overflow areas a day after President Barack Obama visited Minneapolis to tout his federal gun-control proposals.

“For a whole host of reasons, we’re not keeping guns out of the hands of individuals whoshouldn’t have them,” said Dennis Flaherty, executive director of the Minnesota Police and Peace Officers Association. He was specifically backing a bill to require background checks for all gun purchases in Minnesota, cracking down on the sale of guns at shows, online or through unlicensed, private dealers.

That was just one of a dozen proposed limits on firearm ownership under review in the hearings in front of the House Public Safety Committee. The first day drew hundreds of gun owners and activists sitting alongside activists seeking tougher rules on who gets guns. While gun rights supporters wore buttons that read “Self Defense is a Human Right,” activists on the other side had stickers that read, “Minnesotans Against Being Shot.”

With Democrats controlling the Legislature, new limits on gun access have their best shot at the Capitol in a number of years. But the debate could expose divides between urban Democrats, who represent areas where new gun limits are popular, and rural Democrats from areas with high gun ownership and less support for serious curtailments on the ability to own weapons.

“The way to stop gun violence is with another gun,” said Rep. David Dill, a Democrat whose sprawling district includes most of northeastern Minnesota’s Arrowhead Region.

Dill said he’s on the same page with the National Rifle Association in its opposition to the background check bill and most of the other gun proposals, and believes he has enough votes in the House to defeat changes in gun laws opposed by the NRA.

Gov. Mark Dayton, also a Democrat, has not wholeheartedly embraced new gun control measures; he told the Star Tribune on Monday that any changes would need support from rural lawmakers in order to get his signature.

The House committee chairman, St. Paul Democrat Michael Paymar, said he intends to assemble the best state-level proposals into an umbrella bill likely to be dubbed the “Gun Violence Prevention Act.” He said the House is likely to vote on the package later in February.

The gun law push is less active in the Senate, also led by Democrats. Senate Majority Leader Tom Bakk said votes on any gun legislation would take a back seat to the Legislature’s budget.

Besides the background check proposal, lawmakers on Tuesday reviewed a bill to let local police departments order a mental health evaluation on people who apply for the state permit that’s required to carry a weapon. Other gun bills include proposals to ban the sale of assault-style automatic rifles and of certain types of ammunition clips.

Paymar wouldn’t say which among the dozen proposals he thought should definitely be part of the final gun bill. Heather Martens, executive director of the gun control advocacy group Protect Minnesota, predicted the background check provision would almost certainly be part of the package.

“We’re going to make some real progress this session,” Martens said. She said people directly affected by gun violence understand the need for law changes, pointing to testimony Tuesday by the 17-year-old son of the Minneapolis business owner killed in last fall’s fatal shootings at Accent Signage Systems.

“His American dream became an American nightmare,” Sami Rahamim said of his father. Rahamim did not back specific proposals but spoke more generally of the need to tighten access to guns.

The background check bill would require all gun purchases in Minnesota to be made with federally licensed dealers. NRA lobbyist Christopher Rager said sales records of those purchases would eventually be turned over to federal law enforcement, creating a functional registry of all gun owners.

The NRA also raised objections to tougher mental health screening for gun permit applicants, and it found an ally in representatives from the National Alliance for the Mentally Ill. Sue Abderholden, director of the alliance’s Minnesota chapter, said that would stigmatize the vast majority of mentally ill people who are not dangerous and discourage some people who need treatment from voluntarily entering programs out of fear it could weaken future gun ownership rights or just violate their privacy.

Rep. Tony Cornish, a Republican and NRA ally, had planned a bill allowing armed teachers in Minnesota schools as a preventive step against school shootings. While he criticized the Democratic bills up for debate Tuesday, Cornish said he decided not to introduce that measure as a separate bill since Dayton vowed to veto it.

But Cornish said he might resurrect his proposal as a House floor amendment later in the session. A colorful and sometimes brash supporter of gun rights, Cornish showed up at Tuesday’s hearing wearing a tie with the NRA logo and a lapel pin shaped like an AK-47.

“I don’t see anything happening with gun control this year,” Cornish said.