It seems like there may be a confusion between querying "/me" in the context of the app versus the user. The documentation is not very clear on this point, but I have encountered the same issue.
Have you tried utilizing FB.getLoginStatus?
Once I have set up FB.init, I usually make a call to FB.getLoginStatus as shown below. While there might be alternative methods, this approach has been effective for me.
I hope this information is useful:
FB.getLoginStatus(function (response) {
if (response.session) {
// User is logged in and connected, someone within your network
graphUrl = "https://graph.facebook.com/me?access_token=" + response.session.access_token + "&callback=displayUser"
var script = document.createElement("script");
script.src = graphUrl;
document.body.appendChild(script);
} else {
// No active user session, unidentified individual
}});
function displayUser(user) {
alert(user.name);
}