Could you provide an example please?

when you first add the date watcher activated twice
http://codepen.io/PavelCSS/pen/obzeOz?editors=101

I don't think that I've understand yourissue very well. To be honest it seems to me that the watch callback lacks in logic.

Your original pen shows that as soon as the input got focus the alert get a +1. But that's because the watch is missing some logic.

Try this, a simple and ghetto solution, not the best one:

<div id="demo">
  <p>{{alert}}</p>
  <p>Old Val: {{ old }}</p>
  <p>New Val: {{ new }}</p>
  <hr />
  <input id="datepicker" v-model="message" lazy>
</div>

  var demo = new Vue({
    el: '#demo',
    data: {
      message: null,
      alert : 0,
      old: 0,
      new: 0,
    },
    watch : {
      'message' : function(newVal, oldVal) {
        if(oldVal !== null)
        {
          this.alert = this.alert + 1;
          this.old = oldVal;
          this.new = newVal;
        }
      }
    }
  });

$(function() {
  $( "#datepicker" ).datepicker();
});

Code Pen for Reference

It does not work if the value will be written or inserted manually

@PavelCSS It was a silly solution to point out that the problem is how you are implementing the message callback