After the Soviet Union dissolution, the countries independent from the Union, or the countries in the East Europe that were puppets of the Soviet Union turned to the current government without a civil war like most of the nations wanting to change their ideology. Those country also have their own military; it is possible for them to use the military power to stay in power. But it seems nobody did so, except Eastern Germany which used its army to suppress the riots. (Actually, I don’t know a lot about that period, maybe there are some more). And what makes me more confused is none of those countries stayed in communism. (Does that mean the Soviet Union might have a incomptent propaganda department?)
The reason is that these regimes had largely lost support in all parts of the population. By the late 1980s it was obvious that workers in Western Europe had a substantially higher living standard than people in Eastern Europe, and that turned the whole idea of socialism somewhat on its head. Add economical and social stagnation and all the contradictions that arise in any normal society but can not well be addressed in a dictatorship. Even the CPSU had realized that things could not go on this way and appointed a reformer, Gorbachev, as their General Secretary in 1985.
This was a situation in which the loyalty of the army - at least that of the conscripts - would have been quite questionable when deployed against their own people. What can happen when parts of the army turn against their commanders can be observed in Syria post-2011. Even if we assume that the consequences in Eastern Europe might have been less extreme (say, only like in Moscow in August 1991), the leaders were probably aware that deploying the army would not improve outcomes.
In any case, in most Eastern European countries the pattern was that old hardline communist leaders were replaced with more reformist communist leaders, because it was felt that the hardliners had no support in the population anymore (e.g. Honecker by Krenz in East Germany, Husák to Jakeš to Urbánek in Czechoslovakia, Zhivkov to Mladenov in Bulgaria). The reformists then realized that they could not win back popular support and basically gave up - because returning power to the hardliners and calling in the army seemed (to them) a worse idea than giving power to the opposition. So there probably were not many leaders inclined to use the army against the protests in the first place.
The question currently states that East Germany used its army against rioters. This is probably a confusion with Romania, where there were armed confrontations between Securitate (a/the secret police) and protesters and later also between Securitate and the army IIRC. Or a confusion with China (incidentically, East Germany was one of very few countries who endorsed the violent repression of the Tiananmen square protests)
East Germany had confrontations between protesters and police and Stasi (East German secret police) in Dresden, Berlin and Plauen and a few other cities and towns in early October. My impression is that the army played only a minor or no role in these events.