How to check if user is logged in or not with "Google Sign In" (OAuth 2.0)

You do not need to store anything on local storage. The library allows you to check if the user is logged in or not using the isSignedIn.get() on the auth2 of the gapi object.

Load the JavaScript library, make sure you are not deferring the load :

<script src="https://apis.google.com/js/platform.js"></script>

Then initialize the library and check if the user is logged in or not

var auth2;
var googleUser; // The current user

gapi.load('auth2', function(){
    auth2 = gapi.auth2.init({
        client_id: 'your-app-id.apps.googleusercontent.com'
    });
    auth2.attachClickHandler('signin-button', {}, onSuccess, onFailure);

    auth2.isSignedIn.listen(signinChanged);
    auth2.currentUser.listen(userChanged); // This is what you use to listen for user changes
});  

var signinChanged = function (val) {
    console.log('Signin state changed to ', val);
};

var onSuccess = function(user) {
    console.log('Signed in as ' + user.getBasicProfile().getName());
    // Redirect somewhere
};

var onFailure = function(error) {
    console.log(error);
};

function signOut() {
    auth2.signOut().then(function () {
        console.log('User signed out.');
    });
}        

var userChanged = function (user) {
    if(user.getId()){
      // Do something here
    }
};

Don't forget to change the app id

You can stringify a custom userEntity object and store it in sessionStorage where you can check it anytime you load a new page. I have not tested the following but it should work (doing something similar with WebAPI tokens in the same way)

function onSignIn(googleUser) 
{
  var profile = googleUser.getBasicProfile();
  console.log('ID: ' + profile.getId()); 
  console.log('Name: ' + profile.getName());
  console.log('Image URL: ' + profile.getImageUrl());
  console.log('Email: ' + profile.getEmail());
  
  var myUserEntity = {};
  myUserEntity.Id = profile.getId();
  myUserEntity.Name = profile.getName();
  
  //Store the entity object in sessionStorage where it will be accessible from all pages of your site.
  sessionStorage.setItem('myUserEntity',JSON.stringify(myUserEntity));

  alert(profile.getName());   
}

function checkIfLoggedIn()
{
  if(sessionStorage.getItem('myUserEntity') == null){
    //Redirect to login page, no user entity available in sessionStorage
    window.location.href='Login.html';
  } else {
    //User already logged in
    var userEntity = {};
    userEntity = JSON.parse(sessionStorage.getItem('myUserEntity'));
    ...
    DoWhatever();
  }
}

function logout()
{
  //Don't forget to clear sessionStorage when user logs out
  sessionStorage.clear();
}

Of course, you can have some internal checks if the sessionStorage object is tampered with. This approach should work with modern browsers like Chrome and Firefox.