Yes, the brewing process produces some CO2, but not enough for the bubbly head people expect from beer these days.
150 years ago, when people had beer buckets that they'd get filled up and drink from all day, carbonation was much lower. Home brewers talk about it a lot, I'm told.
Considering all the talk about carbon sequestration, you'd think they'd have a way to produce food quality CO2 while cleaning the air. Maybe that'll be the next technological miracle.