No, they didn't. You see, just as both Koreas do today, both Germanies considered themselves as representing the true German nation in the beginning. After 1963, the German Democratic Republic (GDR, East Germany) corrected her course slightly, aiming to solidify the status quo first, and gaining international recognition as a "second Germany" with varying success. The Federal Republic (FRG, West G.) however never officially changed her mind (and had no incentive to do so) and continued to consider the GDR as an unlawful and criminal puppet regime occupying part of the German territory. Thus, all East Germans were considered citizens of the FRG in theory, only prevented by the eastern regime to exert their rights. If you, as an East German, managed to escape from the GDR, West German authorities greeted you with open arms and you would get citizenship.

Furthermore, in the 1980s, the FRG frequently redeemed people imprisoned in the GDR, tried to organize family reunions, mostly through "bribing" the East German authorities with monetary compensation. West Germany also successfully lobbied and paid for the removal of East German spring guns along the German-German border.

On a side note: Keep in mind, that the German-German border was not only heavily guarded and fortified in Berlin but everywhere, even in the countryside. It is however the infamous Berlin Wall which became the most prominent part of the East German border fortifications and the world-wide seen icon of the so called Iron Curtain.