I hope no one gets angry, but I'm going to offer a dissenting voice.
There is the primer anvil with priming compound between the anvil and the primer cup. It takes a lot of force to punch out a primer and the de-capping pin has a small diameter. Therefore a lot of force per unit of area will be applied and there is no way that the anvil will not be deformed and priming compound not displaced. I don't know if that will significantly affect the sensitivity and reliability of the primer. Everyone's experience says that it doesn't, and certainly experience trumps conjecture.
Regardless, I would not use recycled primers on any ammunition that might be used for self defense or hunting and instead limit the recycled primers to range and plinking. But that's just me.