No, it’s entirely self hosted. WireGuard is native to the Linux kernel which is kind of awesome.

Unless you specify otherwise, your traffic to the internet goes out of the default gateway that whatever client machine was using before, eg your home router. But if you configure your client a certain way you can force all outbound traffic from that machine to go through another on your WireGuard network, making it similar to an exit node on a Tailscale network. There are tons of things you can do with WireGuard!