I'm coming at this from finance, not linguistics.
If you buy (partial) ownership when you invest (so buying stock instead of making a loan), you give over the right to distribute the returns made from your investment to the manager of the investment. They do not have to release those returns back to you in full or on a regular basis.
So, since this is their discretion, I think "dividends are payed out" not "dividends are paid out".
Here's another thought, based on this and other sites. There seem to be three appropriate uses of payed:
1) I payed out rope.
2) I payed out caulk.
3) I payed a visit [got this from another site].
I think the common thread there is that each involves some sort of attachment to something larger:
1) I payed out (some) rope (from a larger spool).
2) I payed out (some) caulk (from a larger tube).
3) I payed (some time to make) a visit (from my larger pool of free time).
If I'm correct, then my above answer that the correct form is "payed out dividends" because dividends come from a larger pool of profits earned from the investment.