If I set the max-width property for mdl-layout__header-row class, the dropdowns are not working properly, they are misaligned. Remove the max-width property and everything works fine.
Code:
<!doctype html>
<html>
<head>
<meta charset="utf-8">
<meta http-equiv="X-UA-Compatible" content="IE=edge">
<meta name="viewport" content="width=device-width, initial-scale=1, maximum-scale=1, user-scalable=no">
<title>Test Page</title>
<link rel="stylesheet" href="https://fonts.googleapis.com/icon?family=Material+Icons">
<link rel="stylesheet" href="https://code.getmdl.io/1.3.0/material.red-light_blue.min.css" />
<script src="https://code.getmdl.io/1.3.0/material.min.js"></script>
</head>
<body>
<header class="mdl-layout__header">
<div class="mdl-layout__header-row" style="max-width: 1200px;width: 100%;margin: auto;">
<a href="/">
<img style="height: 60px;" src="http://www.vwshops.com/dummy-store-1/img/my-shop-logo-1452186152.jpg" />
</a>
<span>Logo</span>
<button id="DROPDOWN_RIGHT" class="mdl-button mdl-js-button mdl-button--icon">
<i class="material-icons">arrow_drop_down</i>
</button>
<ul class="mdl-menu mdl-menu--bottom-right mdl-js-menu mdl-js-ripple-effect language-menu" for="DROPDOWN_RIGHT">
<li class="mdl-menu__item">One</li>
<li class="mdl-menu__item">Two</li>
<li class="mdl-menu__item">Three</li>
</ul>
<button style="margin-left: auto;" id="DROPDOWN_LEFT" class="mdl-button mdl-js-button mdl-button--icon">
<i class="material-icons">arrow_drop_down</i>
</button>
<ul class="mdl-menu mdl-menu--bottom-right mdl-js-menu mdl-js-ripple-effect language-menu" for="DROPDOWN_LEFT">
<li class="mdl-menu__item">One</li>
<li class="mdl-menu__item">Two</li>
<li class="mdl-menu__item">Three</li>
</ul>
</div>
</header>
</body>
</html>
Add position:relative. This stops the positioned elements straying out of line in .mdl-layout__header-row div. When elements are positioned using absolute positioning you need to setup a parent container to position them off and do that by setting a position:relative for them - otherwise it will be to the outermost html element / document they position off.
<!doctype html>
<html>
<head>
<meta charset="utf-8">
<meta http-equiv="X-UA-Compatible" content="IE=edge">
<meta name="viewport" content="width=device-width, initial-scale=1, maximum-scale=1, user-scalable=no">
<title>Test Page</title>
<link rel="stylesheet" href="https://fonts.googleapis.com/icon?family=Material+Icons">
<link rel="stylesheet" href="https://code.getmdl.io/1.3.0/material.red-light_blue.min.css" />
<script src="https://code.getmdl.io/1.3.0/material.min.js"></script>
</head>
<body>
<header class="mdl-layout__header">
<div class="mdl-layout__header-row" style="max-width: 1200px;width: 100%;margin: auto; position:relative;">
<a href="/">
<img style="height: 60px;" src="http://www.vwshops.com/dummy-store-1/img/my-shop-logo-1452186152.jpg" />
</a>
<span>Logo</span>
<button id="DROPDOWN_RIGHT" class="mdl-button mdl-js-button mdl-button--icon">
<i class="material-icons">arrow_drop_down</i>
</button>
<ul class="mdl-menu mdl-menu--bottom-right mdl-js-menu mdl-js-ripple-effect language-menu" for="DROPDOWN_RIGHT">
<li class="mdl-menu__item">One</li>
<li class="mdl-menu__item">Two</li>
<li class="mdl-menu__item">Three</li>
</ul>
<button style="margin-left: auto;" id="DROPDOWN_LEFT" class="mdl-button mdl-js-button mdl-button--icon">
<i class="material-icons">arrow_drop_down</i>
</button>
<ul class="mdl-menu mdl-menu--bottom-right mdl-js-menu mdl-js-ripple-effect language-menu" for="DROPDOWN_LEFT">
<li class="mdl-menu__item">One</li>
<li class="mdl-menu__item">Two</li>
<li class="mdl-menu__item">Three</li>
</ul>
</div>
</header>
</body>
</html>