Can I access the shadow DOM using jQuery?

No, not outside of the Polymer element.

After reading up on Polymer, it looks like you can only have access to the shadow-DOM of Polymer elements in scripts within the Polymer element. The Polymer docs on Automatic node finding say:

Every node in a component’s shadow DOM that is tagged with an id attribute is automatically referenced in the component’s this.$ hash.

This means you can add a <script> tag as a sibling to <template> where this.$.test will be the element you want.

<polymer-element name="my-component">
  <template>
    <div id='test'>CONTENT</div>
  </template>
  <script>
    Polymer('my-component', {
        logNameValue: function () {
            console.log('polymer element', this.$.test);
            console.log('jQuery wrapper of polymer element', $(this.$.test));
        }
    });
  </script>
</polymer-element>

You can use $('body /deep/ your-selector') pattern to pierce through shadow DOM and get Jquery to work inside it.

update: So far I have only managed to make this work on chrome for desktop. I believe, other browsers do not support the /deep/ combinator.

update 2: /deep/ combinator is deprecated and should not be used anymore. It is scheduled to be removed from Chrome.