But what both ignore is a well-enshrined legal principle that says no treaty can override the Constitution or U.S. laws.
In fact, a draft of the treaty circulated in New York this week has been criticized by arms control activists for containing too many loopholes. For instance, it doesn't include a proposed ban on ammunition trade. Gun activists are standing firm in near-blanket opposition to such a ban, as last Friday's deadly Colorado theater rampage puts pressure on the issue.
A later draft closed some of those loopholes and negotiators hoped to reach a final agreement by a Friday deadline for action.
While the treaty controversy is simmering in Congress and on the Internet, it hasn't yet become a burning issue in the presidential race.
President Barack Obama supports the treaty effort but hasn't talked about it on the campaign trail. Presumptive Republican nominee Mitt Romney hasn't specifically addressed the treaty but broadly opposes what he sees as overreach by the U.N. on many fronts.
"I'm willing to talk there. I'm not willing to give the United Nations sovereignty in any way or form" over U.S. citizens or law, Romney said July 18.
Romney and Obama did touch on the issue of U.S. gun control laws this week, with Obama suggesting stiffer regulations in a speech Wednesday night to the National Urban League and Romney arguing in an NBC interview from London, where he is traveling, that America does not need new gun laws.
The Constitution's Second Amendment offers broad protection for weapons ownership by civilians. As recently as 2008, the Supreme Court affirmed it when it struck down a ban on handguns in the District of Columbia, ruling that individuals have a constitutional right to keep guns for self-defense and other purposes.
The court also has ruled separately that treaty obligations may not infringe on individual constitutional protections and rights within U.S. borders. This goes back at least to a 1920 ruling that a migratory bird treaty with Canada, which prohibited the hunting or capturing of certain birds, was an unconstitutional interference with states' rights under the 10th Amendment.
Treaties are government-to-government agreements and do not subject citizens of one nation to laws of another or to those of an outside body.
Also, the U.N. resolution that authorized drafting of the small arms treaty recognizes the clear-cut right of nations "to regulate internal transfers of arms" and says nothing in the treaty that emerges will affect "constitutional protections on private ownership" of firearms.
Beyond that, there are many court rulings spelling out the limits of treaties. And if an act of Congress is inconsistent with a treaty obligation, the law passed by Congress prevails. Legal scholars say this has been well established.
A proposed treaty to regulate exports and imports of small weapons has been on the U.N. agenda since 2006, and Bush ordered a U.S. veto of that move.
Obama got the process rolling again in 2010. So far, 152 nations have participated in the drafting, and Secretary of State Hillary Rodham Clinton has pledged to push for Senate ratification once there is a final document.
But it may be an empty gesture.
Treaties must be ratified by two-thirds of the 100-member Senate, or 67 votes. And with pressure mounting from the gun lobby, led by the politically powerful NRA, a bipartisan letter opposing such a treaty already gained the signatures of well over 50 senators.
Even so, fears have spread on the Internet and on social networks lately, with some pro-gun activists suggesting the Obama administration was capitalizing on the Colorado killings to advance its case for gun control and others portraying it as a darker plot by the U.N. to expand its reach.
The controversy feeds into suggestions by many conservatives that Obama ultimately hopes to ban possession of firearms, even though he has stood up for protecting Second Amendment rights.
Some gun rights advocates acknowledge that a treaty by itself wouldn't likely undercut these Second Amendment guarantees.
"But there are all kinds of ways that international law insinuates itself into U.S. law even when there's not a formal ratification," said David Kopel, research director for the conservative Independence Institute.
For instance, he suggests a treaty could affect the shipment of certain gun parts to U.S. manufacturers, even if the United States does not sign the treaty.
But Gabor Rona, international legal director of Human Rights First, said, "The circle created by the treaty and the circle created by the Second Amendment simply don't intersect at all."
"There is no doubt that the Constitution is superior to any international treaties," said Rona, who also teaches international law at Columbia University.