Persisting data locally can be done via AsyncStorage, which is simply a container for key-value pairs of string, string. So you'll use a key (hopefully prefixed by the app id and logged in user id) and a JSON.stringifyed object. Pull it out and JSON.parse it and you're good to go.
If you're using Redux, the library redux-persist helps facilitate that. If you want a custom implementation, you can write your own redux middleware (which is pretty easy to do).
If you're using MobX, check out Mobx State Tree.
Persisting data remotely is not the domain of React Native. It will interact with an API (unless you're interrogating a remote database directly).