Although not a direct answer, this thread with responses from /u/commiespaceinvader and myself will likely be of interest, as it details the extent of Finnish involvement, which was mostly overlooked by historians for a very long time.

Nazi anti-semitism was open and blatant, and anyone who could read newspapers must have been perfectly aware of it by 1939.

However, the Nazis tried to keep the Final Solution a secret from the world, and until concentration camps were liberated, succeeded reasonably well. Even as late as in 1944, even the highest political leadership of the Western Allies apparently tended to disbelieve the rumors and scant eyewitness reports of systematic programme of genocide, and while stories of Nazi atrocities were a regular feature in Allied newspapers and some of these stories focused on mass killings of Jews, it seems that many people tended to take these as exaggerations or outright propaganda.

The information deficit was even worse in Finland, for the obvious reason that anti-German articles were not that popular, might end up in censorship, and were not generally believed. There were rumours about what was happening in Germany and its occupied territories, and Swedish newspapers - which were accessible in Finland - did discuss some of these in greater detail. Nevertheless, they were unsubstantiated at the time, and I believe it was simply hard for people to believe that millions of people would be simply murdered and cremated in an industrial fashion. Ordinary people could easily believe that there were brutalities and even mass killings, but the sheer audacity of systematically killing off an entire segment of the society was just incomprehensibly evil.

Obviously, the small Jewish community in Finland had more reason to hear and believe the rumors, and many Jewish veterans (a total of 267 men and women of Jewish faith served during the war, and about 25 of them were killed in action) have stated that they had heard rumors of atrocities in Germany from their Central European relatives or in their community. They have also stated that this did cause some soul-searching, but in the end, they saw themselves to be fighting for Finland and not for Germany. After all, the Russians were also known to be at least occasionally anti-semitic, and before 1945, the most famous recent outrages against the Jews had been the Russian pogroms of late 1800s. The Finnish Jews were also to a large extent proprietors of their own businesses or educated professionals, and therefore had every reason to fear what would happen to them as "class enemies" if the Soviets overran Finnish defences; and, perhaps most importantly, they felt that they had to prove their loyalty to the Finnish state, which had only extended full civil rights to the Jews after Finnish independence in 1917 - the last European country to do so.

The Finnish Jews certainly distrusted Nazis and Germans generally. There are accounts of Jewish shopkeepers refusing to serve German soldiers on leave, and one field synagogue was set up in the vicinity of the German encampment by the men of Infantry Regiment 24, probably at least partially for the purposes of sending a message of "up yours!" - this with full approval of the Finnish Army's high command. On the other hand, one famous Jewish military surgeon saved the lives of many Germans as well, considering that his dislike of the Nazi Reich could not override his Hippocratic oath. This surgeon, one Finnish-Jewish captain and one nurse serving alongside German forces received German Iron Crosses; all of them refused the award, at least in one occasion in no uncertain words as to where the officer presenting the award could stuff it.

So, in summary, Finnish Jews had certainly heard rumors of mass murders and illtreatment, and had no doubt they would be mistreated if they fell to German hands, but it was hard for even them to believe that even Nazis would actually try to systematically kill all the Jews. It is also worthwhile to understand that the Jews were not the only ones in Finland who were at best ambivalent about Germans: large sections of the populace and political elite viewed Nazis with a mixture of fear and outright contempt. Social Democrats in particular had no illusions about what would probably happen if Germany emerged victorious from the war, and had every reason to expect being eventually hounded to extinction (even if not systematically murdered) if that happened. Nevertheless, Social Democrats sat in the government for the duration of the war.

Main sources here are interviews of Jewish veterans published by Finnish Broadcasting Company (Yle).