The paper-dialog shows up behind drawer-panel when launched from a custom element.
I'm not sure exactly what your requirements are, but I was able to use your dialog element and display it above and not obscured by the panel by simply placing it outside of the <paper-drawer-panel>
.
Here's what it looks like:
I'm not sure if this is what you want, but to do this, something like the following illustrates a basic example:
<paper-drawer-panel>
<paper-header-panel drawer>
<paper-toolbar>
<div>Drawer</div>
</paper-toolbar>
<paper-menu selected="{{_selected}}">
<paper-item>Item 1</paper-item>
</paper-menu>
</paper-header-panel>
<paper-header-panel mode="standard" main>
<paper-toolbar>
<h1>[[_selected]]</h1>
</paper-toolbar>
<neon-animated-pages selected="{{_selected}}">
<paper-button raised on-tap="showDialog">Show Dialog</paper-button>
<div>Div</div>
</neon-animated-pages>
</paper-header-panel>
</paper-drawer-panel>
<my-greeting id="dialog"></my-greeting>