Yeah, I'm not buying that Peter has two distinct Powers... either he copies the abilities of others, or he has his own ability. Peter's Dreams were always AT Simone's Dad's place or, the exception, when he died, a testament to Dad's empathic dreaming abilities, and not Peter's. If Peter ever has any empathic dreams again, I'll recant, but I'm betting he can't anymore.
While I don't believe that everyone should have powers, there is evidence to suggest that Dad did. Otherwise, why would he get up and shout that "Save the World" stuff if HE didn't have powers, and furthermore, have aspects of his dream which Peter distinctly didn't have. Peter copies, Dad has the empathy.
While Peter's connection to people may allow him to hold powers (My foot hovered this morning) such as connecting to Dad's last dream, I don't think Peter is the empath... otherwise, every character deserves a "second mutation" and that would suck.