How do rights/contract work in case of anthologies?

As I understand it, the "normal" workflow is, as the author you get an advance from a publisher, and when the book is published and revenue exceeds the advance, you get the royalties.

You mention you sign the contract with the authors and pay them, which sounds like a one-time deal.

If it's not confidential, how does this work, and how do original publishers (if any) figure into this?

Good question, original publishers (the magazines, in this case) set the terms for how long an exclusivity period they want to purchase for first print rights. It's usually a year, but many magazine contracts have a carve-out for "year's best" anthologies like mine.

I deal directly with the authors, not the original publishers. I pay a fixed rate for reprint rights, which is a one-time deal. This is mostly because pro rata royalties on an anthology are a pain to calculate (I've done this before), but also because it's unlikely for me to make back the money I spend on anthology creation, so if there were ever royalties they would be tiny in any case.

There's not much money in short fiction -- many authors tend to use it as a stepping stone to novels, or just as a hobby.