I am building a Web App and a separate API (so that users can share their collected data with someone if they want to) using Ruby on Rails. The users can log in on the web a
When you say Web app server and a separate API server, which needs to talk to each other every time there is an update from a user on your Web app server. All I can suggest you to break them down to 3 entities as rails engine.
Why Core? Because, when you need to update your business logic, it will be just one place: Core Engine.
Now to answer your question further on authenticating API call from your web app server. You need to:
Once you're done with securing API, you can implement the authentication logic in your Web application. You can use OAuth2 for authenticating your app from API.
Also, to make your API available only to OAuth calls using doorkeeper: https://doorkeeper-provider.herokuapp.com/#client-applications
P.S.: I prefer json response from the APIs, it's a preferred trend I'd say. ;)
EDIT- postman is a chrome extension for making experimental/fake APIs before you actually write them for your application. It's much faster because you'd know what you finally have to design at the end of the day.