No, the extermination of the Jews was intended to be a strictly secret matter and the participants were expected not to speak of it. This comes straight from the horse's mouth; here's what Heinrich Himmler said to a group of high-ranking SS leaders at a conference in Posen in October 1943:

"I also want to refer here very frankly to a very difficult matter. We can now very openly talk about this among ourselves, and yet we will never discuss this publicly...I am talking about the "Jewish evacuation": the extermination of the Jewish people [die Ausrottung des jüdischen Volkes]. It is one of those things that is easily said...But none has observed it, endured it. Most of you here know what it means when 100 corpses lie next to each other, when there are 500 or when there are 1,000. To have endured this and at the same time to have remained a decent person—with exceptions due to human weaknesses—has made us tough, and is a glorious chapter that has not and will not be spoken of."

When Himmler was speaking of the "extermination" of the Jews, the term he used was "Ausrottung". Notably, eight months earlier, when Goebbels gave his famous "total war" speech at the Sportpalast in Berlin, said this, barely avoiding a slip of the tongue:

"Germany, in any case, has no intention of bowing to the Jewish threat, but rather of confronting it in due time, if need be in terms of the complete and most radical ex...suppression [Aus...schaltung]."

He nearly said "Ausrottung", but caught himself and changed it to "Ausschaltung", which literally means "turning off", but in this case means something like "suppression", a more vague term than the unambiguous "Ausrottung".

That said, Hitler himself had publicly referred explicitly to the "annihilation" of the Jews in his speech to the Reichstag on 30 January 1939, where he famously said:

"Once again I will be a prophet: should the international Jewish financiers succeed, both within and beyond Europe, in plunging mankind into yet another world war, then the result will not be a Bolshevization of the earth and thus the victory of Jewry, but the annihilation of the Jewish race in Europe [die Vernichtung der jüdischen Rasse in Europa]."

Hitler used the term "Vernichtung" (annihilation) rather than "Ausrottung" (extermination), but again, the meaning is unambiguous. This so-called "prophecy" was repeatedly referred to in the coming years, both in Nazi propaganda and by Hitler himself. It was also acknowledged by the outside world. The implications were obvious.

I wrote an answer recently about what ordinary Germans knew about the fate of the Jews if you're curious about the details of that question, but the tl;dr version is that anyone who wanted to know could figure out what was happening. As Frank Bajohr and Dieter Pohl put it, the mass murder of the Jews was an "open secret" among the German public.

As far as the fate of the population in the areas of Poland and the Soviet Union that were intended to be part of the Greater Germanic Reich, Nazi propaganda was frank about the elimination of the native inhabitants of the territories that were to be colonized by Germanic settlers. The Nazis undertook an intense propaganda campaign to justify to the German public the necessity of eliminating the Jewish and Slavic Untermenschen so that Germanic peoples could expand eastward. This idea was an integral part of their indoctrination of Hitler Youth members and even schoolchildren, through materials such as a pamphlet entitled "On the German People and its Territory" (Vom deutschen Volk und Seinem Lebensraum). Any reasonably intelligent person should have been aware of the implications of Nazi population policy and could have figured out that their plans for the settlement of the East would have required the elimination of tens of millions of people.

Sources:

Ian Kershaw, Hitler, the Germans, and the Final Solution (Yale UP, 2008)

Richard Evans, The Third Reich at War (Penguin, 2008)

Stephen G. Fritz, Ostkrieg: Hitler's War of Extermination in the East (UP of Kentucky, 2011)

Himmler's Posen Speech (excerpted; the relevant text is on page 7)

I have previously found relevant answers by /u/commiespaceinvader and /u/estherke in To what extent was Hitler able to keep the brutality of the concentration camps secret from the Allies and the German people?

See also At what point did the existence of Nazi concentration camps become public knowledge and what was the public response? answered by /u/PeculiarLeah.