I don't peel them. I never have and my mother never has. We have a specific "mushroom brush" that we bought in Germany, that is supposed to be designed for brushing soil off mushrooms but all it does it make them go soggy so I don't use that either. Instead I dampen a piece of kitchen paper and wipe them very gently to remove all the dirt.
I can't think what benefits there would be to peeling them. I'm not aware of French or Italian chefs peeling mushrooms and if they don't see the need I can't imagine why anyone else would!