did they ever say whose aunt she was? Since everyone called her "Aunt Bea".

She was Andy's aunt but not Barney's although he and & Andy were first cousins.

http://www.tagsrwc.com/faq/index.php?cat=11#a52

Aunt Bee was Andy's father's sister, because they have the same last name. Since Barney's last name is different, but he still calls Bee "Aunt", his mother was probably their sister, as well?

Not quite. We learn in "Aunt Bee's Invisible Beau" that Aunt Bee is not Barney's aunt. He says, "Boy, for the life of me, I can't figure you out, Andy. If Aunt Bee was my aunt, I wouldn't be staring out of windows. I'd get in the car, I'd drive straight over to Mt. Pilot, I'd get that Hendricks out in the alley and POW!" In other episodes, Barney refers to his mother's side of the family as if Andy does not know any of them. An example is in "Cousin Virgil," when Barney says, "You're really gonna like my cousin Virgil. He comes from my mother's side of the family; they're the fun ones." So, the correct relationship must be: Andy's mother is the sister of Barney's father (assuming that they are FIRST cousins.)

I thought so because in the first episode she says that Opie looks like Andy did at that age; but then how come everyone called her Aunt Bea?