I can see why you might think this but a raster is a raster and it is always rectangular. Setting a 'nodata' value doesn't remove those pixels. The pixels are still present but transparent for display and excluded in many processing tools. Those pixels are also in the same data type (integer, float, etc) as the other "good' or 'on' pixels since you can only store pixels with one data type.
The only real way to reduce the data size is to clip the image to a smaller size (subset data from ROI) or to use a data type that requires less storage space (int or float converted to byte). If you have ROIs that are scattered throughout the image, then your image will only be reduced (clipped) by the *smallest rectangle boundary that encompasses all ROIs*, so the size may not be reduced by that much at all.