One, it seems, can't let the other lob an insult without chucking back a bigger one. It's getting hard to tell who has the thinner skin.
Kim took the bait and responded directly to Trump, firing off his own verbal barrage calling Trump a "mentally deranged US Dotard
," adding that "a frightened dog barks louder".
The icing on Kim's confection of caustic commentary was his threat he'll "tame" Trump "with fire" -- a throw down exposing the hollowness of Trump's threat to North Korea last month.
Back in August, responding to Kim's scary long-distance ICBM rocket tests that put the US in range of his weapons, Trump told Kim that if he continues, he'll be met with "fire and fury like the world has never seen."
Two days later Trump doubled down on this, saying he hadn't been tough enough and that the US military is "locked and loaded".
If there wasn't a thermonuclear trigger lurking in the background, the bombast might be easier to shrug off with the school yard adage: "Sticks and stone may break my bones, but names will never hurt me". But here on the Korean Peninsular, people are worried.
Reports in the Washington Post
that Kim's henchmen are reaching out to senior Republicans to figure Trump out only hint at the depth of concern.
South of the DMZ (de-militarized zone) in Seoul they are used to Kim's -- and his father and grandfather's -- irrational rantings. They have learned to live with this particular style of brinksmanship on their border.
But now Trump is puncturing the illusion that whatever the rhetoric, conflict is beyond the horizon. In the past month people here have begun asking western journalists what is going on with Trump and where is all this talk actually going to end up.
Despite repeated reassurances from Trump's top team that US policy to North Korea has nothing to do with regime change and is all about curbing his missile and nuclear weapons programs, the President keeps signaling that may not be the case.
On Monday, following Kim's Foreign Ministers speech at UN Trump, tweeted: "If he echoes thoughts of little rocket man, they won't be around much longer." At the minimum, that's a clear threat of regime change.
For Trump and Kim, the UNGA has not produced peacemaking but instead escalated the prospect of a misstep. Both are locked in a cycle of threat and counter threat. Kim is now saying he'll test a hydrogen-tipped ICBM over the Pacific.
But Kim wasn't the only leader to react to Trump's UNGA rhetoric. Iran's leaders began new missile tests after hearing Trump rail on them, threatening to unpick their nuclear deal with the West: "That deal is an embarrassment to the United States, and I don't think you've heard the last of it -- believe me."
Indeed Trump's ability to Tweet and speak his way in to a firestorm seems unrivaled at the moment. Days after the UNGA he picked a hugely divisive fight with the NFL and many others over patriotism and the national anthem.
With so much distraction, Kim sent an envoy to Moscow, Russia undoubtedly relishing the prospect of messing with Trump on another global issue and scoring yet more international influence.
Some will have been reminded of when Kim's grandfather, Kim Il-sung, went to meet Stalin to get his backing for starting the 1953 Korean War. He lost to overwhelming US and South Korean force and ultimately only survived with China's help.
Kim's grandfather was irrational in the face of obviously overwhelming odds, and was smashed because of it. Yet he claimed victory.
If Kim understands history as we do and as South Korean's do, not on what his Grandfather built his dynasty on, then even Trump's most provocative challenges can be swallowed.
But if he was brainwashed at his grandfather's knee the way his citizens are by him, then he will be weighing the cost of war to his people against the price of saving face.
For sure, unlike Trump, he has no one advising him to ratchet down his rhetoric and seems to be pushing himself to the only logical conclusion he can see: to perfect his nuclear deterrent.
Short of a war that leaves Trump light on options, the obvious path Trump should be looking to take here is to tone down his diatribes and stop winding Kim up with name calling and threats of regime change.
The question being be asked in South Korea is can he really do it. They've lived with the uncertainty of one outspoken leader for decades. But two tangling together? That's a far more dangerous ball game.