I want to store sensitive data locally in a React Native app.
Is the data only available to the app that wrote it?
For very sensitive app or user data, you could try something like https://github.com/oblador/react-native-keychain on iOS(uses iOS Keychain) or https://github.com/classapp/react-native-sensitive-info for both Android and iOS(uses Android Shared Preference and iOS Keychain).
Both of them come with very fluent API and straightforward way of linking with react-native link
and are a more secure way of preserving data you want to keep away from prying eyes.