Lighting for shots such as this one is not about absolute lighting levels, it is about lighting ratios. The 'dark' parts of the picture don't need to be in absolute darkness. They just need to be less bright than the bright parts of the scene by enough to make them dark by controlling exposure.

As long as the side you want illuminated is lit brightly enough compared to the side you don't want to be bright and exposure is set appropriately you'll get the effect you are looking for.

In both cases when you want the background to be bright or when you want it to be dark the key is to get enough distance between your subject and the background so that light illuminating the parts of the scene you want to be bright does not spill onto the parts you want to be darker. The same is true of walls off to camera left and right: they need to be far enough away and non-reflective enough that light won't bounce off of them and back into the scene.

With a dark background you can also flag your off camera flash to reduce light spilling onto the background. When a bright background is required then the you have fewer options to control the additional light aimed at the background. Distance is the primary way you have of controlling the light coming off the background. Making sure you're only using the minimum power needed on your light illuminating the background in order to blow out the background (instead of, say, twice as much as you need) is another. Tilting the background towards the light source if it is off axis and away from your primary subject may also help.

For how this works out with a dark background, please see: If I want to shoot darker shots, is it better to increase the lighting and then darken in post to retain clarity? Note that the dark areas in the photo can be much brighter than they look in the image, exposure just has to be set to 'kill the ambient' and the parts you want to be bright have to be lit brightly enough to still be bright at that reduced exposure value.

For another question that deals with a lighter background, please see: How to get this particular black and white look?