How to authenticate Firebase on localhost through already auth-ed host

firebase-auth-manual-indexdb.js

var db;
function createDatabase() {
    const request = window.indexedDB.open('firebaseLocalStorageDb', 1);
    request.onsuccess = (e) => {
        // Create DB connection
        db = request.result;
    };
}
createDatabase();

function addData(data){
    const transaction = db.transaction('firebaseLocalStorage', 'readwrite');
    transaction.oncomplete = function(event) {
        //...
    };
    transaction.onerror = function(event) {
      //...
    };
    const objectStore = transaction.objectStore('firebaseLocalStorage', {keyPath: 'fbase_key'});
    // Add login
    const request = objectStore.put({fbase_key:data.fbase_key, value: data.value});
    // Get
    // const request = objectStore.get("firebase:authUser:abcdefghj:[DEFAULT]");
    request.onsuccess = () => {
        console.log(`new data added:`, request.result);
    }
    request.onerror = (err)=> {
        console.error(`Error to add: ${err}`)
    }
}


function getData(fbaseKey){
    const transaction = db.transaction('firebaseLocalStorage', 'readwrite');
    transaction.oncomplete = function(event) {
        //...
    };
    transaction.onerror = function(event) {
      //...
    };
    const objectStore = transaction.objectStore('firebaseLocalStorage', {keyPath: 'fbase_key'});
    // Add login
    //const request = objectStore.put({fbase_key:data.fbase_key, value: data.value});
    // Get
    const request = objectStore.get(fbaseKey);
    request.onsuccess = () => {
        console.log(`GET:`, request.result);
    }
    request.onerror = (err)=> {
        console.error(`Error to get: ${err}`)
    }
}


// On your logged host..
const data = getData('KEY_ON_HOST')

// On your localhost
addData({
    fbase_key: 'KEY_ON_LOCAL',
    value: data
})