Rep. Mark Baker of Brandon and Sen. Will Longwitz of Madison say they want the change because a newspaper in New York published names and other information about people there who have concealed weapons permits.
The two Republicans said permit holders shouldn't have to worry about their address or other personal information being publicly disclosed.
No permit is required for people to have guns in their homes, businesses or vehicles, and no permit is required for antique or collectible weapons. Baker said the document gives permission for someone to carry a pistol or revolver concealed in a bag or under a jacket in other places, such as public sidewalks. A background check is required before a permit is issued.
Baker told The Associated Press that he filed a bill after receiving requests from constituents who believe it's none of anyone else's business if they carry concealed guns.
"It just struck me that simply because the document is in possession of the government does not, in my mind, make it a public document," Baker said.
Under current law, information about a concealed weapons permit becomes public record 45 days after the document is issued or the application is denied. The documents are filed with the state Department of Public Safety.
Mississippi Press Association president Jim Prince said he supports Second Amendment gun rights but opposes putting restrictions on public records. He said the current law works.
"While I sometimes wish we could enact laws to prevent newspapers from behaving irresponsibly, we can't. Trampling the First Amendment to protect the Second is a slippery slope," Prince said in the Madison County Journal, which is published by his company, Prince Newspapers.
Longwitz said in a written statement to the Madison County Journal that there's a balancing act between public safety and the First Amendment rights of free speech.
"Anti-gun activists have already promised to go state-to-state 'outing gun owners,'" Longwitz said. "Making conceal-carry information public puts law-abiding Mississippians in danger."
Attorneys for the Legislature are still drafting bills that have been requested this year, and Longwitz's bill did not show up on the legislative website Friday.
Baker's is House Bill 149.
Baker said he did not file the bill at the request of the National Rifle Association. He also said he does not have a concealed weapons permit.
It's been rare in recent years for Mississippi newspapers to publish the identities of people with concealed weapons permits.
The Associated Press asked Baker whether closing access to concealed weapons permits might make it more difficult for people to find out if an angry neighbor or potential stalker has state permission to carry a concealed weapon.
"I think if a person is in a position where they feel someone is threatening to commit a crime against them, they should communicate with the law enforcement agency where this is occurring or where they reside," Baker said.
Under proposals by Baker and Longwitz, law enforcement officers would still have access to information about concealed weapons permits.
Follow Emily Wagster Pettus on Twitter: http://twitter.com/EWagsterPettus