Most styles of snowshoes are symmetrical, and will work on either foot, but some bindings are designed to be right or left handed, meaning that they are easier to operate with one hand or the other.
Lace up bindings can be worn on either foot without any problem, but snowshoes equipped with buckles and straps on one side of the binding are easier to operate on the respective feet they are designated for (ie. right-handed buckles on the right foot, left-handed buckles on the left foot), which puts the buckles and straps to the outside of the feet.
There are also however some styles of snowshoes that are asymmetrical, specifically engineered to provide clearance over the adjacent shoe, these shoes are meant to be worn on specific feet:
Left Foot Running/speed snowshoe:
Ultimately, your snowshoes will still work for you regardless of which foot you put them on, but they will work better, and you'll have an easier time putting them on and walking if you put them on the feet they're setup to be on.