Proof of Stake - Where are the nodes?

I'm currently researching proof of stake and trying to wrap my head around the whole concept, and one thing that I have yet to be able to find information on is...where are these "nodes" that are actually doing the computation to approve a transaction? For proof of work mining rigs are used to validate and process transactions. With proof of stake though there is no mining, you simply stake coins and the larger your stake the higher your chance of being selected to approve the next block (or did I misunderstand that).

Currently I have staked a small amount of crypto in a proof of stake project and am receiving payouts at set intervals. But I don't know how, as my machines are all turned off during some of these payouts...so it's not my machine that's approving the transactions...it has to be another "node" somewhere else right? If that is the case are these just random servers setup on the network? If so who's paying for these "nodes"? Would there be any benefit in running one yourself?