I'm not a historian, but I wrote a paper on how East Germany dealt with the legacy of fascism while I studied abroad in Berlin, so I've dusted that off. I can't speak to how W. Germany dealt with it, though, so I'll give half the picture and I'm sure someone else will come along with the other half. (Jesus Christ this became a lot longer than I anticipated, sorry in advance.)

The DDR (E. Germany), to begin with, had a problem of establishing state legitimacy that the BRD (W. Germany) didn't really have. To gain this legitimacy, the DDR essentially built up a "foundation myth" of sorts entirely centered on the history of communism in Germany. The liberation of the concentration camps (especially Buchenwald) was attributed to communist-led resistance within the camps. The promulgated narrative was of German communists leading ethnically and politically diverse bands of antifascists from within Germany itself to conquer fascism.

So, in tune with this narrative, after the workers' uprising in June 1953, which attested to the shaky legitimacy of the Communist regime, Buchenwald was turned into a national memorial, and a memorial was erected to Ernst Thaelmann, the murdered leader of the Weimar-era Communist Party of Germany, at the Buchenwald crematorium.

Like Jews were forced to wear a gold star, communist victims in the Holocaust were forced to wear a red triangle, which became a symbol of heroic resistance in art. Socialist realist novelists like Marchwitza, Bredel, and Gotsche told stories spanning generations, constructing a specifically German socialist tradition, progressing from defeat to ultimate salvation by the Red Army.

East German historians used Marxist linearity and "historical materialism" to link the Communist Party to uprisings in German history like the medieval Peasants' War and the 1848 revolution. Marxist thought considers socialism the natural result of revolution stemming from the oppression of the proletariat inherent in capitalist structures, so Marxist East German historians re-interpreted fascism as the last gasp of the elites to protect their interests from the laborers. With this re-interpretation, antifascism and anticapitalism became more or less synonymous. The Berlin Wall was characterized as the "antifascist protective wall."

German monopolists were blamed for instigating WWII, and the non-resisting majority were not accounted for. WWII was a story of the superior social structures of the socialists inevitably defeating imperial fascists, and the DDR really pushed the narrative that it had always been communist and was the first victim of Nazi aggression. Non-communist Nazi resistance, the racial aspect of Nazism, and the mass collaboration allowing for the SS terror system went largely unmentioned. Mass exterminations were largely left out of the narrative as well, because DDR historians were only allowed to understand Nazism in the economic sense of state monopoly capitalism. Non-political victims of the Third Reich were by and large ignored.

Because of the "scientific" inevitability of Marxist history, individuals were largely absolved from blame, and the lesson taken from the rise of fascism was something akin to "don't be capitalist." As (at least according to the narrative) antifascists, East Germany considered itself a co-victor in WWII. My paper has a quote from a DDR student saying "When I imagined WWII as a child, it was as if everyone had somehow been a member of the White Rose or had met secretively in back courtyards and basements to organize resistance and print pamphlets."

Naturally, considering the official historical narrative, many former Nazis made careers in the East German regime. Arno von Lenski, for instance, was an assessor of the Volksgerichthof in Nazi Germany, in a capacity where his signature was appended to multiple death sentences, but in the DDR, Ulbricht awarded him the "Medal for Fighters against Fascism 1933-45." Ernst Grossmann was an SS guard at Sachsenhausen, which largely housed political prisoners, and served on the Central Committee of SED in the DDR, and was decorated as a "Hero of Work." These are two of many examples of the blind eye turned towards the contributions of individual collaborators in the Nazi regime.

On committees such as the Opfer des Faschismus (Victims of Fascism) and the Union of Persecuted of the Nazi Regime (VVN), both party-sanctioned groups, non-political victims were originally included, but by the late '40s were already marginalized. Even non-communist political victims, like the supporters of the July 20 coup, were excluded from the VVN in 1949.

Entire concentration camps were forgotten in the official record, like Marzahn, which held Sinti and Roma but not communists. It took until the 1980s until DDR historians could discuss Jewish persecution and bourgeois antifascist opposition, but even in 1984 lesbians were arrested for visiting Ravensbrueck, the site of persecutions of homosexuals. The Volkskammer did not accept German responsibility for the Holocaust until April 12, 1990.

Sources:

Brinks, J.H. "Poilitcal Anti-Fascism in the German Democratic Republic." Journal of Contemporary History 32.2 (1997): 207-17.

Diner, Dan. "On the Ideology of Antifascism." Trans. Christian Gundermann. New German Critique 67 (1996): 123-32.

Hell, Julia. "At the Center an Absence: Foundationalist Narratives of the GDR and the Legitimatory Discourse of Antifascism." Monatshefte 84.1 (1992): 23-45.

Jurausch, Konrad H. "The Failure of East German Antifascism: Some Ironies of History as Politics." German Studies Review 14.1 (1991): 84-102.

Monteath, Peter. "Narratives of Fascism in the GDR: Buchenwald and the 'Myth of Antifascism'." The European Legacy 4.1 (1999): 99-112.

Plum, Catherine. "The Children of Antifascism: Exploring Young Historians Clubs in the GDR." German Politics & Society 26.1 (2008): 1-28.

Timm, Angelika. "The Burdened Relationship between the GDR and Israel." Jewish Claims against East Germany: Moral Obligations and Pragmatic Policy. Budapest: Central European UP, 1997: 166-180.

TL;DR East Germany created a foundation myth regarding antifascism and socialism which promulgated the narrative that East Germany was socialist and had always been socialist, backed by Marxist historical materialism. The myth essentially turned East Germans from promoters, collaborators, and silent toleraters of Nazism to victims of fascism, at the expense of pretty much every non-communist victim of the Nazi regime.

East Germany, as a matter of political theory, displaced blame for the atrocities of the Nazis onto West Germany. Fascism was characterized as a bourgeois government form, which had naturally been eradicated by the rise of Socialism in the East, but was still alive in the bourgeois West. There's a lot more theory here, but that's the gist. Transferring the blame was made easier by the fact that most Eastern politicians were Communists from before WWII, and had thus been persecuted by the Nazis.

This is in contrast with West Germany, where after 'denazification," many former Nazis were allowed to return to positions of power. This was used in propaganda by the GDR (East) against the FRG (West).

On the other hand, official commemoration of the Holocaust didn't really happen in the East. Monuments listed the dead by country, but didn't differentiate ethnic Poles from Polish Jews, for example.

Ossies also focused on WWII as a class struggle. In Sachsenhausen, the closest concentration camp to Berlin, there is a huge granite obelisk, with red granite triangles on it. Red triangles were given to political prisoners in the camps--mostly Communists. (Pink triangles went to homosexuals, black to criminals. Communist Jews wore a red triangle and a yellow together to form a star. There are more triangles, as well.) So, rather than tell the story of the mostly Jewish victims of the camp, they focused on the Communist martyrs. There's a huge interpretive building outside the camp as well, with big Socialist Realist stained glass works telling how the Communists fought gloriously against the Nazis.

In the West, the crimes of the Nazis weren't really grappled with until the 1960s, when young people who had come of age after WWII started poking around and being outraged at some of the history behind the people in charge. In the 1940s and 50s, the control of the Western Allies over the FRG was more visible (the occupation of Germany only came to a legal end in 1991) and the Allies were worried about creating a strong, democratic, allied state more than they were worried about justice for past crimes. It was easier to cooperate with former Nazis than to wipe them out of the picture. (after all, almost everyone with experience in running the country for more than a dozen years was a Nazi).

As the Cold War dragged on, Eastern Europe became a haven for former Nazis--because the East used the former Nazis in the West as a propaganda tool, they didn't want to find former Nazis in their own backyards, and so didn't cooperate with trying or extraditing them. On the other hand, West Germany continued to grapple with its Nazi roots, and understand, atone for and come to terms with the horrors of the past in a way that East Germany did not, and, I would argue, that was based on the openness allowed by a democratic form of government, as opposed to a dictatorship.

I hope this answers some of your questions.

Not an expert but as there's no real answer yet:

West Germany: a lot of Nazis became part of the system, judges and politicians (not the elite, but party members of the NSDAP). Sorting them out was a major point of the youth movements in '68 and the development of the "German Fall" , the Rote Armee Fraktion (RAF). The young Germans basically blamed their parents to be old Nazis and holding onto the old system. Though the allies forced the German population to look at the horrors of the concentration camp and the third Reich (also the Nuremberg trials made it clear for the German public what had happened) . So no one publicly stood by the old times, though a lot of people kept their copy of Mein Kampf. The Germans also quickly forgot the war because the economy was booming and life post war was good. German Gemütlichkeit and Heimatsgefühl (longing or feeling for home, kinda like patriotism but I wouldn't call it that cause it has a different meaning in Germany) developed in movies and music (Volksmusik, Schlager). The youth began to listen to rock n roll and rolling stones, and setting themselves apart of the elder generation. In Germany it was next to rebellion also distancing from the old times. Real reconciliation started when the generation, which was born after the war or was to young, gained more and more social and political power (~70s).

East Germany (less sure about this): I think the Soviets portrayed west Germany as 'old Nazis' , and used it for their propaganda: west Germany equals evil. As far as I know real reconciliation never occured, but of course there were people from the SS/NSDAP living in the eastern zone, which was ignored by the regime.

I don't know if it helped but that's what I recall from history class in (Western, but unified) Germany.

Edit: the part with the movies

In terms of academia:

Each side had their own historiographic interpretation of Nazi crimes all the way up to the the end of the Cold War (and even today to some extent). Really your question hits the nail on the head: how does each side "deal with the legacy"?

In the West (i.e. for West German, American, British, and French historians for the most part), the preferred mode was a 'functionalist' interpretation. The idea is that those who committed the terrible crimes of WWII did so as a result of mitigating factors other than their simply being evil. For example, a functionalist might say that the terrible, murderous crimes of Nazi Germany were committed as a result of the power struggles within the Nazi bureaucracy or leadership, or as a result of careerism. This was a convenient concept, because many in the West German government had been participants in the Nazi state, and they were key allies of the US.

In the East (the USSR), the method was 'intentionalism'. This sees the organizers of the Holocaust, or Gestapo agents, etc... as totally evil people who gained power alongside other psychopaths. It's much more accusatory and direct, which is quite telling considering the experiences of the USSR during WWII. This is a 'top down' interpretation which sees the Nazi leadership as directly responsible for organizing master plans of genocide or aggression since even the 1920's.

In the last twenty years or so, the two methods have begun to come together in a very productive way. After the end of the Cold War, the scholarship started to interact, and many histories now style themselves as a 'synthesis' of the two.