Anyone close enough to wonder why it was suddenly so windy and blinding were told the explosions were being set off at a safe distance. For instance, Area 51, the army base in the middle of the Nevada desert (where conspiracy theorists believe the Army is reverse-engineering UFOs), was actually one of the most active nuclear test sites in the world. Russia was able to set off their weapons in the similarly desolate region of the country known as "the part that's not Moscow."
And letting the fallout get blown off to the part that's not Russia.
But as technology advanced and the bombs grew bigger and more explode-y, the idea that there was such a thing as a "safe distance" was rendered ridiculous. For instance, according to the Seattle Times, "over the years the atmospheric tests conducted over America exposed a quarter-million assembled troops, plus communities downwind in Nevada and Utah, to an estimated 12 billion curies of radiation, or 148 times the release from the 1986 Chernobyl nuclear-plant meltdown."
Being the more pragmatic of the superpowers, in 1961 the Soviets decided to get all of their reckless endangerment out of the way with one test -- the Tsar Bomba, thusly named because of the Soviet tendency to put "tsar" in front of anything that's stupidly big.
The Tsar Tank was to regular tanks what Ferris wheels are to regular tanks.