Note
Did a bit more reading on the 1978-79 Beagle Channel dispute. It seems that Argentina's military made plans (which became public after the fall of the dictatorship) for a full-scale invasion of Chile - meanwhile inviting Peru and Bolivia, both of whom had century-old grievances of their own with Chile, to join the party. They apparently thought they could take Santiago in a few days (which if nothing else demonstrates that 1982 wasn't the first time they made a highly ambitious assessment of their own military capabilities).
One of the Argentine generals of the time created quite a stir many years later by saying that if the war had actually happened (it's still debated whether any troops actually crossed the border before the invasion was called off), Chile would have won it. Apart from anything else, one imagines the US would have got involved (if only to restore the status quo ante); while the Pinochet regime had become something of an embarrassment to its erstwhile American backers by 1978 (their blowing up Allende's foreign minister in a Washington DC street was unhelpful in this respect), and in any case under Jimmy Carter the US was starting to get out of the business of backing any dictator who could demonstrate a sufficient detestation of Commies, a general war in South America certainly wouldn't have been in US interests.
All that said, the Argentine grievance was legitimate - Chilean control of the disputed islands and their territorial waters left the port of Ushuaia without usable access to the open sea. (The settlement eventually brokered by the Pope involved Chile retaining the islands but the waters being declared international).
These days the two countries seem to get on fine (except in football). The old rule that democracies never go to war with each other (or the variation, that two countries with a McDonald's never go to war with each other) was a casualty of the various 1990s conflicts in the former Yugoslavia, but still holds more often than not, and certainly all the heat went out of this dispute once the two countries became democratic. (Actually, it's exceedingly rare these days for two countries to go to war with each other anywhere these days; conflict takes other forms).
Some further reading (this time, on the webpage of the international campaign to ban land mines) revealed that, after some initial foot-dragging (possibly because deep down they didn't lose too much sleep over the odd splattered smuggler), Chile is about halfway through the task of clearing its old minefields, though it's expected to take them until 2020 or thereabouts to finish the job. I'm guessing that Tierra del Fuego, where they don't pose much of a threat to anything except the odd errant sheep, is a fair way down the priority list.