This is far from comprehensive, but gives a general idea that some high-profile defections did occur. Most of the defections by Germans from the FRG to the GDR were connected to Stasi intelligence. For example, Hans Joachim-Tiedge was one of the famous cases from the 1980s and his rationale for defecting is still nebulous.

But there were other cases of defection that had little to do with espionage. Manfred von Brauchitsch defected to the GDR to escape his tax debts and the GDR offered him financial incentives to do so. Even more enigmatic is the case of Otto John who defected to the GDR in 1954 partly out of the conviction that West German rearmament and the continued presence of Nazis within West German society ran counter to the ideal that German after 1945 had turned its back on National Socialism.

John's case highlights an important point about the Cold War in Germany, especially in its early decades. Although it is tempting to conceptualize the two Germanies as "democracy vs. totalitarianism," the distinction was less clear cut during the time. Although Adenauer's West Germany made public commitments to democracy, it possessed a high degree of social conservatism and one of the pillars of its political society was a fervent anticommunism. For many Germans that rejected Nazism, the latter was particularly troubling given that the FRG adapted a very lenient stance towards denazification. Adenauer cemented a silent compact that offered former Nazis (at least those who were not directly responsible for major crimes) a place in the new government provided that they kept their mouth shut about the previous regime and did not directly undermine democratic institutions. This was something that fed a great deal of ideological dissatisfaction among a number of West Germans. Although things like the 1953 East Berlin uprising and the Berlin Wall soiled the image of the GDR and communism, a reaction against the social and political strictures of the Adenauer state was a consistent theme within the FRG throughout the Cold War.

Sources

Frei, Norbert. Adenauer's Germany and the Nazi Past: The Politics of Amnesty and Integration. New York: Columbia University Press, 2002.

Kundnani, Hans. Utopia or Auschwitz?: Germany's 1968 Generation and the Holocaust. New York: Columbia University Press, 2009.

I don't know of anyone who permanently left West Germany to go east, but at least some members of the Red Army Faction used East Germany as a safe haven to evade the authorities in West Germany. It wasn't political, just practical - East Germany wouldn't cooperate with West Germany's desire to pursue them as criminals, and East German authorities apparently liked the trouble they stirred up enough to turn a blind eye to them.