There is certainly no rule against a river in the rain shadow. Rain shadows are not necessarily dry, per se, they just tend to be drier than the opposite side. An east coast rain shadow (at a latitude where weather patterns travel west to east) will tend be wetter than a west coast rain shadow at the same latitude, as storms coming in from the warm water east coast will tend to carry more moisture to the rain shadowed area, and in some cases bring in large amounts of tropical moisture from north moving tropical storms like hurricanes on the east coast of the US. Rivers through deserts are certainly possible, see the Nile, Colorado River and Yellow River, so I doubt there is a reason not to be able to have snow melt create a river in even a desert rain shadow. To pass through a hot desert more than seasonally would require a heck of a source, I suspect, to defeat the evaporation.