Do creatures who fail a saving throw, know they failed a saving throw? I imagine they don't, but I failed to find a specific rule saying so.
Not inherently, no. Though when you fail against say a fireball or lightning bolt being fried/electrocuted full in the face is pretty obvious. Charm person? Not so much.
You do know whenyou MAKE a saving throw however: I'd imagine its like feeling a bird hit a window on your brain.
Do creatures who fail a saving throw, know they failed a saving throw? I imagine they don't, but I failed to find a specific rule saying so.
Succeeding on a Saving Throw: "A creature that successfully saves against a spell that has no obvious physical effects feels a hostile force or a tingle, but cannot deduce the exact nature of the attack. Likewise, if a creature's saving throw succeeds against a targeted spell, you sense that the spell has failed. You do not sense when creatures succeed on saves against effect and area spells."
As for failure, I suppose it depends on the spell.
You shouldn't realize you just failed to save to disbelieve an illusion, for example.