Is this difference tied to a screen size?
Maybe. Depending on the screen size and the screen space available it'll be better to show the menu on the left, on the right, on top of something, moving the content to one direction and then showing the menu in that place (the side menu, on mobile, are represented using a hamburguer icon a lot of times, take a look: http://pttrns.com/categories/1-activity-feeds).
Or is it because designers implement them without much thought to consistency?
I believe that the consistency involved in this component is that it opens a menu that is a list, like "here is a list of what you can do", how to open it is a different matter, and I don't believe there would be a default behavior regarding direction since we use the component on very different situations.
Is it just a preference?
It definitely happens. As you are designing, there are aspects about function that you will think about, like the screen size, available screen space, the information presented on the screen, but there will be more subjective things like your taste or aesthetics.
What else could influence this decision?
Should the menu be fixed? If that is the case, you probably don't want it to sit on top of the content, maybe you will want to open it from left to right, pushing the content to the right and reducing the space available for it.
Where is the menu located?
Is it on the right? Maybe you will want to open the menu from the right to left, so when it is opened, the cursor will be already on the menu area or very close to it. Look at your example from squarespace.com, after you click the menu, the cursor will be right on top of stories! You don't need to move the mouse too much, seems efficient to me.