Esri has a useful blog post on utilizing Python's multiprocessing module. In many cases the multiprocessing module works fine with Esri's arcpy site package, mostly for embarrassingly parallel geoprocessing tasks that do not require any shared data (e.g. feature classes in a file geodatabase) among the workers.
I have run into so many problems developing arcpy based multiprocessing applications that I rarely bother any more and usually either focus on streamlining and optimizing existing code or opting for open-source multiprocessing solutions.
In any case, when using the multiprocessing module, you will need to optimize the multiprocessing on your machine as there will be a threshold where the workers max out your system resources resulting in diminishing returns.