The bomb we today call the "Tsar Bomba" (the moniker is post-Cold War from what I have been able to determine) was not a secret. It was announced by Khrushchev well ahead of its being detonated (it was usually called the "50 megaton bomb" or "100 megaton bomb" at the time), with enough time for the UN to issue a resolution protesting its detonation.

Anyway, the news was that the Soviets had set off a big bomb, with varying estimations about the yield (the Soviets themselves were not totally sure). It was seen as saber rattling by Khrushchev, and denounced as such by most Western leaders. JFK issued a press release from the White House, condemning its fallout (which was a tricky line for Kennedy because the US couldn't denounce fallout too much without committing to not testing such weapons itself), denouncing it as a "military act" that had no scientific purpose (internal assessments by the CIA and laboratories disagreed with that). My favorite lines:

There is no mystery about producing a 50-megaton bomb. Nor is there any technical need for testing such a weapon at full-scale detonation in order to confirm the basic design. The United States Government considered this matter carefully several years ago and concluded that such weapons would not provide an essential military capability.

Which is a very careful way of saying, we could do this too, if we wanted to. And indeed the US did have such options on the table, in its classified quarters. (And the weapon was tested at half-scale — saying you don't need to test a weapon at full-scale is true but irrelevant here. In fact in general the Soviets were much more prone to partial-yield testing than the Americans.)

Popular coverage generally took the "official line" that one big bomb is not as powerful as several smaller-yet-still-large bombs. Life magazine called it "a superbomb more bluff than bang." My favorite response was from the New York Herald Tribune's editorial page, which took a somewhat more skeptical view of the "official line":

Military experts are saying that a 100-megaton bomb such as Mr. Khrushchev talked about — one with a wallop equal to 100 million tons of TNT — would be too big to be efficient. That's nice.

Anyway, that's the gist of it. The issue came back in a minor way with the discussions for the Limited Test Ban Treaty (which would effectively prohibit the US from developing its own weapons of that yield range — a prohibition on atmospheric testing would "lock out" research into "very high yield" weapons), but US officials (McNamara in this instance) asserted that the US just wasn't that interested in super large weapons, that its superiority in quantity and accuracy of weapons, and its advantages in the areas of miniaturization of weapons, was superior to having large, extremely heavy bombs.