maybe add a standard 'share' component from react-native?

https://reactnative.dev/docs/share

and the logic is when you click on pdf or whatever button you have - the share component will be opened and then you'll choose where you want to put that file(also to save it on your phone)

You can use npm i react-native-fs library for save in local devices

Sample Code:

import React from 'react';
import {View, Text, TouchableOpacity} from 'react-native';
import RNFS from 'react-native-fs';

export default function App() {

  //if you want to save in download directory then you can use this path
  const path = RNFS.DownloadDirectoryPath;

  const saveFile = () => {
    RNFS.copyFile(yourCurrentFilePath, yourDestinationFilePath);
  };

  return (
    <View style={{flex: 1}}>
      <TouchableOpacity onPress={() => saveFile()}>
        <Text>Save</Text>
      </TouchableOpacity>
    </View>
  );
}

For More Information you can explore the react-native-fs library documents react-native-fs