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I have two different after-market fuel caps on my C172. The right tank
cap is vented; the left is not. Should both be vented? Is a properly
functioning vented cap one that only allows air to flow into the tank?

You have to very carefully study this. The Cessna fuel systems changed over
the years and you can get into big trouble screwing with the wrong caps.

However, for most of the 172's, the primary vent is always the left tank vent behind
the strut. The right tank is normally vented through a cross feed line that connects
the tanks roughly behind the pilot's head. The right cap has a normally closed vent
that only opens when the right tank gets sufficiently negative pressure due to the crossfeed
being blocked.

That all being said, one of these "normally closed" caps should not cause a problem on
the left side as well, but that's not the way Cessna ships 'em.