With the understanding that it only answers part of your question (I need to see if I can get enough sources together to tackle the other part), I wrote this post about American Jewish knowledge of what was going on, and u/commiespaceinvader wrote a more detailed post about HOW they (and people in the Allied nations in general) found out here.

This is a surprisingly difficult question to answer. First of all, there's a kind of depressing sampling bias--it's possible people who thought that being "rounded up" meant certain death would've been more likely to attempt to hide even at great risk, which would mean that people who survived might be more likely to be people who knew what was happening and believed it.

Second, asking people afterwards is also subject to difficulty. People remembering of their thoughts and feelings will be colored by what they know now. Or they'll try to emotionally justify why they did show up at a deportation with their family, by saying they didn't know what would happen. It could be even at the time they really did know but were in denial.

Also a note--I assume by "concentration camps" you're really talking about the death camps. People often conflate the two, but they were really quite different (though some camps had both).

The only really good source to figure this out is from archival materials from the Warsaw Ghetto. A small group of people (led by Jewish historians) collected an enormous archive in the Warsaw Ghetto. Their goals were both to preserve real-time experiences of what this tumultuous time was like for Polish Jewry, but also to preserve accounts of what life was like before the war. This archive, variously called the Oyneg Shabbos Archive or the Ringelblum Archive, was stored in large metal milk containers underground, and nearly all of it has been recovered. The archivists tried to get as really broad collection of material as was possible, and they included legal documentation, transcriptions of accounts, write-ups about communal life before the war, discussions of how Jewish life continued during the war, correspondence, sermons, poetry, etc. You can see a catalog of it here. Making things more difficult for later historians, the material had to be stored hastily without extensive cataloging, some material was damaged, and a significant amount was written using pseudonyms or codes to conceal personal information in case the archive was discovered. Because of this the full archive was only published recently, and it's enormous and not really so accessible. Smaller collections were published earlier. One such example is "Kiddush hashem : Jewish religious and cultural life in Poland during the holocaust", which is a volume of Rabbi Shimon Huberband's material, focusing on a picture of prewar Jewish life in Poland, and how Jewish life continued in the ghetto (both before and after the ghetto was sealed), along with a smattering of other documentation.

In the achive there are several accounts from people who had escaped death camps, both Chelmno and Treblinka. The archives contain information about the camps compiled from escapees. So the information seems to have reached the Warsaw Ghetto. There's no real way of knowing how widely this information was disseminated, though there are references in Huberband's work to widespread rumors of mass killings.

As time went on and more people were "deported" from the Ghettos without reappearing (people occasionally leaving for forced labor assignment was not unheard of), including whole families who were not good candidates for labor, it would've been increasingly apparent that the reports were accurate. This explains the ghetto uprisings when the ghettos were liquidated--those in the ghetto understood that if they complied with roundups they were very likely to be killed, so they started uprisings even though the revolts were extremely unlikely to succeed.