I disagree. You are mixing apples and oranges.
Democracy is simply a form of government. The laws under which citizens in that democracy live can be determined in any number of ways.
Sharia is sort of a hybrid..a system of immutable laws with instruction on how to govern
Communism is neither. It is a model for an economic system and it is, at is core, silent on law or form of government. Of course, when you start taking things away from people that used to belong to them, you are going to need some oppressive laws and a dictatorial form of government to keep the thing going.