I don't see the logic in after. Doing it before and after both have the benefit of adding water which makes brushing teeth much nicer. However, doing it before also has the benefit of washing away many of the nasty things left on the toothbrush - mainly old toothpaste and germs which multiplied on the brush from your last brushing . If you do it after, you really aren't washing much of that stuff off. Imagine taking a filthy brush, covering the top with paste, and then washing. The paste just blocked the water from doing the cleaning. Yuck. You can do it if you want to, I'll stick to before.
Both is just silly, before does everything you need. So it is just a very minor waste of time and water.