Interesting.
I'm guessing here, but I'd say, apart from some small amount of gas exchange in the nasal membrane blood vessel capillaries (.5%?), the rest of the O2-CO2 exchange occurs in the skin glands, specifically eccrine sweat glands. While the appocrine (stinky scent sweat) glands, associated with hair follicles (eg. arm pits), are one-way-exiting skin glands; the eccrines (unscented salty sweat, mostly in the volar surfaces of the palms and soles but in humans spread throughout the body including the arm pits), are two-way-skin glands.
"Sweat glands draw fluid from the bloodstream and pass the water to the surface. They normally reabsorb salts and recycle them back into the bloodstream".
don't sweat it questions.(Daily Break) | Article from The Virginian Pilot | HighBeam Research
Not sure, but I'd expect that as long as the skin is kept moist, (humans sweat all the time insensibly, even while asleep) O2 and CO2 interchange can take place there at the molecular level (not as visible bubbles). The fact that we have many (2.6m) active eccrines compared to most mammals (and birds and reptiles) might indicate past selection for alternative routes of oxygenation supplementing a somewhat limited lung/blood/muscle O2 holding capacity.
Do we breathe through our skin cells? No. Do we breathe through our sweat glands? AFAICT, Yes, bypassing lungs/gills, we can respire via cutaneous eccrine electrolyte recycling, but only at a very low rate. This is why we are vulnerable to toxins through our skin, they can be absorbed through the eccrine skin glands (not through the apocrine glands, sebaceous glands, hair, nor skin cells).
Feet and noses: they run, they smell and they breathe!
DDeden