<dom-module id="using-listeners-obj">
  <template>
    <style>
      :host{
        width: 220px;
        height: 100px;
        border: 1px solid black;
        display: block;
      }

      #inner{
        width: calc(100% - 10px);
        height: 50px;
        border: 1px solid blue;
        margin: auto;
        margin-top: 15px;
      }
    </style>
    <div>Tap me for alert message</div>
    <div id="inner">Tap me for different alert</div>
  </template>
</dom-module>
<script>
  Polymer({
    is:'using-listeners-obj',

    //Listener Object
    listeners:{
      //tap a special gesture event in Polymer. Its like click event the main difference being it is more mobile deivce friendly
      'tap':'tapped',  //this will get executed on host of the element
      'inner.tap':'divTapped' //this tap will get executed only on the mentioned node (inner in this case)
    },

    tapped:function(){
      alert('you have tapped host element');
    },

    divTapped:function(e){
      event.stopPropagation();  //this is only to stop bubbling of event. If you comment this then tap event will bubble and parent's(host) listener will also get executed.
      console.log(e);
      alert('you have tapped inner div');
    }
  })
</script>

Another way to add an event listener is to use on-event annotation in DOM. Below is an example using up event, which is fired when finger/mouse goes up

<dom-module id="annotated-listener">
  <template>
    <style>
      .inner{
        width: calc(200px);
        height: 50px;
        border: 1px solid blue;
        margin-top: 15px;
      }
    </style>
    <!-- As up is the name of the event annotation will be on-up -->
    <div class="inner" on-up='upEventOccurs'>Tap me for different alert</div>
  </template>
</dom-module>
<script>
  Polymer({
    is:'annotated-listener',
    upEventOccurs:function(e){
      //detail Object in event contains x and y co-ordinate of event
      alert('up event occurs at x:'+e.detail.x+' y:'+e.detail.y);
    }
  })
</script>

You can also add/remove listener imperatively using listen and unlisten method of Polymer

    <dom-module id="imparative-listener">
      <template>
        <style>
          #inner{
            width: 200px;
            height: 50px;
            border: 1px solid blue;
            margin-top: 15px;
          }
        </style>
        <div id="inner">I've imparative listener attached</div>
      </template>
    </dom-module>
    <script>
      Polymer({
        is:'imparative-listener',
        
        //keeping it in attached to make sure elements with their own shadow root are attached
        attached:function(){
          this.listen(this.$.inner,'track','trackDetails'); // For more details on method check https://www.polymer-project.org/1.0/docs/api/Polymer.Base#method-listen
        },
        
        //all the listener functions have event as first parameter and detail (event.detail) as second parameter to the function
        trackDetails:function(e,detail){
          if(detail.state=='end'){
            alert('Track distance x:'+detail.dx+' y:'+detail.dy);// For more details on track event check https://www.polymer-project.org/1.0/docs/devguide/gesture-events
      }
    }
  })
</script>

You can also fire your own events and then listen to them from either Polymer element of HTML page

This Element fires the custom event

    <dom-module id="custom-event">
  <template>
    <style>
      #inner{
        width: 200px;
        height: 50px;
        border: 1px solid blue;
        margin-top: 15px;
      }
    </style>
    <div id="inner" on-tap="firing">I'll fire a custom event</div>
  </template>
</dom-module>
<script>
  Polymer({
    is:'custom-event',
    firing:function(){
      this.fire('my-event',{value:"Yeah! i'm being listened"}) //fire is the method which is use to fire custom events, second parameter can also be null if no data is required
    }
  })
</script>

Here's an element which is listening to that custom event

  <link rel="import" href="custom-event.html">
 <dom-module id="custom-event-listener">
  <template>
    <style></style>
    <custom-event on-my-event="_myListen"></custom-event>
  </template>
</dom-module>
<script>
  Polymer({
    is:'custom-event-listener',
/*    listeners:{ //you can also use listener object instead of on-event attribute
      'my-event':'listen'
    },*/
    _myListen:function(e,detail){
      alert(detail.value+"from Polymer Element");
    },
  })
</script>

Listening from HTML

<html>
  <head>
    <meta charset="UTF-8">
    <title>Events</title>
    <script src='bower_components/webcomponentsjs/webcomponents-lite.min.js'></script>
    <link rel="import" href="./bower_components/polymer/polymer.html">
    <link rel="import" href="custom-event-listener.html">
  </head>
  <body>
    <!--As event bubbles by default we can also listen to event from custom-event-listener instead of custom-event-->
    <custom-event-listener></custom-event-listener>
  </body>
  <script>
    document.querySelector('custom-event-listener').addEventListener('my-event',function(e){
      console.log(e);
      alert(e.detail.value+"from HTML");
    })
  </script>
</html>

Properties with notify:true also fires an event

<link rel="import" href="../bower_components/paper-input/paper-input.html">
<dom-module id="property-change-event">
  <template>
    <style></style>
    <paper-input id="input" label="type here to fire value change event" on-value-changed='changeFired'></paper-input>
  </template>
</dom-module>
<script>
  Polymer({
    is:'property-change-event',
    changeFired:function(e){
      if(e.detail.value!="")
        alert("new value is: "+e.detail.value);
    }
  })
</script>

It is possible to retarget an event in Polymer ie you can change event details like path thus hiding the actual details of an event/element from user.

For e.g if a div inside event-retargeting element is firing the event but developer does not want the user to know that he can retarget the event to event-retargeting element by using the following code.

var targetEl = document.querySelector('event-retargeting');
var normalizedEvent = Polymer.dom(event);
normalizedEvent.rootTarget = targetEl;
normalizedEvent.localTarget =targetEl
normalizedEvent.path = [];
normalizedEvent.path.push(targetEl);
normalizedEvent.path.push(document.querySelector('body'));
normalizedEvent.path.push(document.querySelector('html')); 

To see the working example please refer to plunker in remarks section.