Well, it did indeed happen, but not quiet in the way you seem to think.

People from the west had, unlike the people in the east, the freedom to travel. Including the right to travel the GDR. Which made "escaping" not quiet necessary. After all you could just leave Western Germany without a notice of departure.

In fact, depending on the sources, between 475,000 and 646,000 people emigrated from West to East Germany between 1949 and 1990. The numbers are unclear because we have the sources from the GDR on the one hand, the ones from the West on the other hand. One of the reasons is, that the West didn't include people in their statistics who left for the East without a notice of departure. Furthermore, these numbers were used from both sides for their respective propaganda - while the GDR saw these numbers as a proof for the appeal of the socialist state, West Germany took the numbers of those who escaped for the West as an argument against the GDR. At the same time, both sides were not quiet comfortable with the situation. Both feared that migration could lead to the disclosure of sensible information and therefore weaken the own state. Plus, the GDR had quiet extensive background checks and were not willing to take just anybody.

The question for why is a bit more difficult to answer, of course people had different motivations. There are relatively little written sources from Migrants, but those who did give a reason for why they left for the GDR almost always saw the socialist system as a necessary lesson drawn from the NS history. If we are looking at the reasons officially given - upon registration in the GDR or if you decided to leave West Germany with a notice of departure, you had to explain your reasoning to the officials - we will see that a lot of people named "Family and Friends" as their reason. Other frequently used reasons were economical reasons, not finding an acceptable place to live, homesickness or for marriage.

But since you specifically asked for "escaping", yes, there were also people who escaped to the East. The most prominent example would be members of the RAF (Rote Armee Fraktion). After the German Autumn in 1977 several members of the RAF were on the run. Inge Viett, one of the members, approached the Staatssicherheit who offered her to grant 8 members asylum in the GDR. But even then it was nothing like the people escaping from the east, the eight members simply travelled from West Germany to Prague and then to the GDR.

Source: Bernd Stöver - Zuflucht DDR: Spione und andere Übersiedler, München 2009