A <div> is not a VueComponent, which means it doesn't have an $emit method.
So to make your Vue custom directive emit an event, you will have to do some checking first:
- If the directive was used in a Vue custom component, then call $emit()of that component's instance
- If the directive was used in a regular DOM element (...because there's no $emit()...), then dispatch a native DOM event using.dispatchEvent.
Luckily, Vue's v-on listeners respond to native custom events.
That should be all. Demo implementation below.
Vue.component('my-comp', {
  template: `<input value="click me and check the console" size="40">`
});
Vue.directive('my-directive', {
  bind: function (el, binding, vnode) {
    // say you want to execute your logic when the element is clicked
    el.addEventListener('click', function (e) {
    
      var eventName = 'my-event';
      var eventData = {myData: 'stuff - ' + binding.expression}
      if (vnode.componentInstance) {
       vnode.componentInstance.$emit(eventName, {detail: eventData}); // use {detail:} to be uniform
      } else {
       vnode.elm.dispatchEvent(new CustomEvent(eventName, {detail: eventData}));
      }
    })
  }
})
new Vue({
  el: '#app',
  methods: {
    handleStuff(e) { console.log('my-event received', e.detail); }
  }
})
<script src="https://unpkg.com/vue@2.5.15/dist/vue.min.js"></script>
<div id="app">
  <div v-my-directive.modifier="'native element'" @my-event="handleStuff">CLICK ME AND CHECK THE CONSOLE</div>
  <hr>
  <my-comp v-my-directive.modifier="'custom component'" @my-event="handleStuff"></my-comp>
</div>