How to Reference/Echo Variable from Another PHP Function

As indicated in the comments, you should return the string, not echo it:

add_shortcode('yt', 'getYoutubeDetails');

function getYoutubeDetails($atts) {
    extract(shortcode_atts(array(
    'video' => ''
    ), $atts));

    // Get YouTube data via the API
    $JSON = file_get_contents("https://gdata.youtube.com/feeds/api/videos?q=$video&alt=json");
    $JSON_Data = json_decode($JSON);
    $views = $JSON_Data->{'feed'}->{'entry'}[0]->{'yt$statistics'}->{'viewCount'};
    $views = number_format($views);
    $duration = $JSON_Data->{'feed'}->{'entry'}[0]->{'media$group'}->{'yt$duration'}->{'seconds'};


    $s = "<iframe width=\"1280\" height=\"720\" src=\"//www.youtube.com/embed/$video?showinfo=0&rel=0&modestbranding=1&theme=light&iv_load_policy=3&autohide=1&enablejsapi=1\" frameborder=\"0\" allowfullscreen></iframe>";
    $s .= "<strong>Views:</strong> $views<br><strong>Duration:</strong> ";
    $s .= sec2hms($duration);
    $s .= "<br /><br />";
    return $s;
}