[Sprocket]

I did some quick and dirty Everstone testing in HeartGold. I bred Male and Female Magikarps (since they have the shortest hatch rate). Male had a Brave nature, Female had a Quiet nature.

Test 1: Male holding Everstone, Female holding nothing. 10 eggs.
7 Brave offspring, 3 other offspring.

Test 2: Female holding Everstone, Male holding nothing. 10 eggs.
7 Quiet offspring, 3 other offspring.

Test 3: Both Male and Female holding Everstones. 20 eggs.
8 Quiet offspring, 6 Brave offspring, 6 other offspring.

Admittedly it's a small sample size, and a larger sample size is needed to confirm (hence quick and dirty). But based on these results:

If only one parent holds the Everstone, that parent will be the one to pass down the nature (50% of the time or so).

If both parents hold an Everstone, you (seem to) have about 1/3 chance of passing the male's nature down, 1/3 chance of passing the female's nature down, and 1/3 chance of a random nature.

If you're so inclined, two parents with the same nature and both holding Everstones, you (seem to) have about a 2/3 chance of passing down the nature.

And I believe it was brought up earlier in the thread, but I've also seen a Female + Ditto pairing, with the Female holding the Everstone, pass down the Female's nature and not the Ditto's nature (across 5+ eggs out of 10 or so). This needs further testing as well.