You can use shorter code to check for weekend => date('N', strtotime($date)) >= 6.
So, to check for 2 dates — and not just 1 — use a function to keep your code simple and clean:
$date1 = '2011-01-01' ;
$date2 = '2017-05-26';
if ( check_if_weekend($date1) && check_if_weekend($date2) ) {
    echo 'yes. both are weekends' ;
} else if ( check_if_weekend($date1) || check_if_weekend($date2) ) {
    echo 'no. only one date is a weekend.' ;
} else {
    echo 'no. neither are weekends.' ;
}
function check_if_weekend($date) {
    return (date('N', strtotime($date)) >= 6);
}
Using your existing code, which is slightly longer, following is how you would check for 2 dates:
$date1 = '2011-01-01' ;
$date2 = '2017-05-27';
if ( check_if_weekend_long($date1) && check_if_weekend_long($date2) ) {
    echo 'yes. both are weekends' ;
} else if ( check_if_weekend_long($date1) || check_if_weekend_long($date2) ) {
    echo 'no. only one date is a weekend.' ;
} else {
    echo 'no. neither are weekends.' ;
}
function check_if_weekend_long($date_str) {
    $timestamp = strtotime($date_str);
    $weekday= date("l", $timestamp );
    $normalized_weekday = strtolower($weekday);
    //echo $normalized_weekday ;
    if (($normalized_weekday == "saturday") || ($normalized_weekday == "sunday")) {
        return true;
    } else {
        return false;
    }
}