What I noticed is that when we communicate with pictures, we communicate through a symbolic sort of shorthand. Instead of drawing a real life human, it is easier to draw a stick figure. Instead of literally drawing lotion, we draw a man with a sunburn and a bottle - with the hopes that someone will guess "sun tan lotion" and therefore finding the word lotion.
This is not only true of "Draw My Thing" but also of games like Taboo. In such games, we tap into a cultural reference bank and think of other ways to arrive at words without actually describing the word directly (since such directness is either complicated - i.e. too hard to draw, or taboo - i.e. against the game rules)
On the converse side, I also draw what some people would consider "artistic picture". In one of my classes, I don't have a notebook and rely on some friends to lend me a piece or two per class. Then, I tax myself and draw them a picture to make up for my "burden". Usually my drawings might be of cars, or buildings. But what is interesting is that the picture of the cars that I draw in this class are more realistic and full of detail and size proportions, then my simple two dimensional, profile of a car with wheels that sort of resembles a punch buggy.
My point is: in different situations, we have different ideas to communicate. Often when we have time constraints or other limitations (rules or ability) we get the point across as best we can. Moreover, the our culture helps create and maintain symbols that we can utilize in order to more effectively get our idea conveyed to another person.
Overall, I think it is an interesting angle to think about communication.