Not to deter further discussion, but there's a good answer by /u/estherke here.

There's a lot more information available in the FAQ section on the Holocaust and Nazi Crimes Against Humanity.

Estherke's answer is on point and I agree with their characterization entirely. I would add a few things pertinent to your particular quesiton:

1. Governments outside of Eastern Europe were aware of "rumors" of what was going on in Eastern Europe with the concentration camps and different ranking members of government believed the claims to different degrees. It wasn't until late in the war (as estherke points out) that the Allies were certain of what was taking place. Let's remember that even the people being packed up and forced into these camps were lead to believe they weren't headed for their deaths and the Nazi's did their best to make sure that the evidence of their actions was as limited as possible.
2. It wasn't so much foreign governments that aimed to keep how concentration camps operated a secret but rather domestic governments who tried to hide it from international governments. Most governments were aware that Jews and other religious and ethnic minorities were treated poorly under Nazi and Nazi-aligned governments and it was known that prison and work camps had been organized, but as estherke points out there was a big difference between these "work" camps that you found in Germany and the extermination camps that were prevalent Eastern Europe. It should also be noted that the concentration camps played a limited role in the carrying out of the Holocaust and governments were much more aware of things like the Einsatzgruppen and their actions due to the fact that these acts were done in a much more public setting.

TLDR: There were rumors of these camps that were slowly substantiated, the rapidity of the political and military situation in Europe lead to confusion. Intentional obscuration by Nazis further confused things, governments not affiliated with the Nazis did very little (if anything) to hide the atrocities that took place in the camps.

I hope my rambling answers your question.

Works consulted: FDR and the Holocaust - Great for information related to the US governments thought process on the Holocaust ('What did the President know and when did he know it' kinda stuff) The Rise and Fall of the Third Reich - Talks a lot about the Einsatzgruppen and their role in the Holocaust as well as their practice in the field. There's also a good chunk on the Nazis consciousness of the Holocaust. Accounts of the Wannsee Conference - They knew they had to be less upfront about what they were doing and how they were doing it. Hitler's Willing Executioners - Perhaps a little inflammatory, cited here for sake of completeness, it does a good job making arguments for who knew what and when in Europe.