How do you implement a P2P payments system?

Hello there, I've been researching peer-to-peer payments. The technical implementation is quite perplexing. For example, consider chipper cash, which offers no-fee p2p payments. How would you go about putting this into action? I saw a video of the cofounders saying that the initial version required two wallets to send payments and that it used bitcoin rails under the hood. So I'm guessing they're using some sort of bitcoin lighting network for such cheap and fast transactions, despite the fact that they're using fiat currencies. I'd like to know how two people can transfer money to each other using a peer-to-peer model, as well as the technical details behind it.