I never saw the word "bribee" before. If I saw it in print I'd guess the meaning correctly.
If I had to describe someone who took bribes I might call that person "corrupt," though that means a person who customarily takes bribes. In a specific case, as Emily says, even if I knew the word "bribee" I wouldn't use it, I'd just use a phrase--like this example from a news story: "authorities didn't question the bribe taker."
"Bribee" is the Scrabble dictionary, and it's in an Android app that's supposed to be the Merriam-Webster Third International, but it's not in my trusty American Heritage desk dictionary. So it's a "real word" but it's not a common one.