Only oxygen if it were freshwater, so you are absolutely correct.And if there aren't two streams of bubbles? Because in my experience, there is invariably only one.
In salt water the 2 "streams" are not H and O, but rather sodium (from the salt) and chlorine (also from salt), because they have lower potentials. The sodium reacts instantly in the presence of water into caustic and hydrogen gas (the bubbles you see). The chlorine rapidly reacts to form hypochlorous acid (bleach) which disolves. Thus, only one stream on the zinc and the slight smell of bleach on the cathode, though certainly too dilute to notice. (This is how Electroscan heads work. This is also how bleach and caustic are made.)