A heavy breather is a generic name for a person that is in a position of power and influence. They do not breathe any louder than the rest of us, but their breath/words have a larger impact on society. This term usually is used reserved corporate executives or high level politicians.
"So dog, whaddid ya do today?! Dude, I had to sit in this meeting with a bunch of heavy breathers"
"The heavy breathers of the right are panting for spending cuts again"