Why a strange discrepency between get_current_user_id() when using AJAX versus output of document.cookie?

I had to trace through WordPress core and the REST API plugin code to fix this, but finally did it. The solution is described here.

The TL;DR of the solution is that the WP-API will ignore the content of the login cookie and assume an unauthenticated (logged out) request unless the correct value is given in the _wpnonce HTTP GET parameter (in the query string) or the HTTP_X_WP_NONCE HTTP header. My JavaScript did not correctly include this. The corrected JavaScript method looks like this:

/**     
 * Checks for any new comments.
 */     
var pollForNewComments = function () {
    var url = api_base + '/comments&post=" + getPostId() + "&offset=" + getCommentCount()
        + "&_wpnonce=" + wpApiSettings.nonce;
    jQuery.get(url, function (response) {
        if (response.length) {
            appendComments(response);
            showNewCommentsNotice();
        }
    });
};

Note that adding the _wpnonce parameter in the query string with the wpApiSettings.nonce global JavaScript variable (which is added by the REST API plugin) fixes the issue for me.