Standard Notes is free and open-source software, which means self-hosting both ends (the app and server) is possible and relatively simple.
The first is the backend infrastructure. The backend is zero-knowledge, which means it does not understand the contents of what it is storing. Any user content received by the server is always encrypted and cannot be understood by the server.
The server is responsible for authentication and syncing. When you self-host your own server, the end result will be an endpoint that you expose via HTTPS. You enter this endpoint in the Standard Notes client applications in the Custom sync server field (found under Advanced options when signing in or registering).
Because the client applications (desktop, web, and mobile) allow the option of specifying which server to connect to, self-hosting the client applications is not as common as self-hosting the server. If you were to go down this route, the paths you might take are: