"Duck and cover" is about reducing the possibility of being injured from flying debris, collapsing ceilings, breaking glass, and, when outside, the amount of thermal radiation deposited directly onto skin. If everybody exposed to a nuclear blast did it, there would be some increase in saved lives, decreased injuries, etc. At some distances from ground zero it would have no likely effect of your survival at all, at some it would not be necessary. But in between is a lot of potential area and potential lives. To think that "Duck and Cover" would mean total survival is to misunderstand it. It is a relatively "cheap" way of decreasing injury if you think an atomic attack is going to happen. For various locations of the "victim," and very types of weapons over the Cold War, its efficacy would vary (the canonical film was made in 1950, when the Soviet nuclear weapons were few in quantity and relatively small in power — things changed, technologically, over the course of the Cold War, and led instead to a focus on fallout mitigation instead of blast survival).

As for the public — that's a harder question to answer. There is a lot of anecdotal evidence that people came away from such training scared, not emboldened. The core idea in the "Duck and Cover" training is that nuclear weapons use is not just hypothetical, but a real threat. (The language was often about when the bomb was dropped, not if the bomb was dropped.) The 1950 film is kind of terrifying when looked at in this lens: it is about teaching children to believe that nuclear weapons could be used at any time, that you needed to be alert and on guard 24 hours a day, that no activity was so innocuous or banal that you should not be ready to take cover. That's a pretty heavy message.

As for whether the general public believed the Civil Defense people — there is a lot of evidence that they didn't think the activities would work or help, a lot of ridicule of the idea that a school desk would hold up against an atomic bomb. There's a lot going on here, including a rejection of expertise and a lack of understanding about what such activities were meant to do.

The real problems with Cold War Civil Defense were not that this kind of thing wouldn't work, though it often gets focused upon. The really dubious plans were things like evacuation, the notion that you could get everybody out of Los Angeles, DC, New York, etc., if you thought an attack was imminent. As the mayor of Los Angeles put it, they couldn't get people out of Los Angeles smoothly on a sunny Friday afternoon at 5pm — what are the odds they'd be able get it done under threat of atomic attack?

A nice book on Civil Defense and its effect on US culture is Kenneth Rose, One Nation Underground: The Fallout Shelter in American Culture (NYU Press, 2004).