So, I will argue "very little": Orchid's decentralized node "directory" runs on Ethereum, but transactions are only very seldom made to modify it (it is more than just servers coming and going, but not much more); in the future, when Ethereum moves to proof of stake, I'd expect this to drop from "very little" to "almost none" (well, as "overhead").

The real blockchain transaction volume we do is for backing the payment system, and there I'll note that Orchid's payment layer is somewhat chain agnostic: I designed the payment protocol to support some level of negotiation on how the payments happen. Right now, many of the active users happen to use xDAI (which is not based on proof of work).