What are two significant challenges the people in Eastern Europe and the former Soviet Union faced immediately after the collapse of communism?
I would argue that the two most significant challenges faced by the people of Eastern Europe were A) setting up a new governmental system and B) setting up a new economic system. Both of these had to be done essentially from scratch after the fall of communism. This is a problem that has not completely been resolved in all of these countries.
After the fall of communism, these countries had to figure out how to maintain order and keep a government running without communism. None of these countries had had any experience of non-communist governments in many decades. This continues to be a problem as some countries still have dictators or strongmen like Vladimir Putin in power and have not really managed to become democratic.
The countries also had to try to form viable economies. This was very difficult since they had little experience with private ownership, competition, or any of the other aspects of modern capitalism. Again, this continues to be a problem as countries such as Russia have what is called "crony capitalism."
Having to create governmental and economic systems from scratch was a major problem for Eastern European countries after communism ended.