Which public? And when during world war 2? And which camps?

Some camps, such as Dachau (near Munich), were already in place long before World War II started. Others, such as Westerbork (in the Northeast Netherlands), were actually created by domestic governments as refugee camps and re-purposed by the Nazis into concentration camps. Then there are others, such as the Operation Reinhardt camps, which were mostly secret during the war, although there was a general understanding of the fact that the Nazis were doing very bad things to Jews in Eastern Europe.

The American government was aware of the existence of the death camps relatively quickly. For example, Operation Reinhardt officially began in the autumn of 1941. The US government was aware of the German policy of genocide by August 1942 at the very latest, as the World Jewish Congress was already sending evidence to the US State Department by that time.

But there is a difference between what people "knew", what the could have known and what they actually believed. Elie Wiesel famously points out in his pseudo-memoir Night that a local Jewish man returned to his village in Hungary telling stories of the horrors of the camps only to have the local Jewish population not believe him. The local Jews were deported to Auschwitz a short time later.

Sources: USHMM, Friedlander 1, and 2