React.js Serverside rendering and Event Handlers

I am learning to use react.js and have some issues to use the event handlers. The final question would be: Is it possible to use server side rendering and send event handlers to the client automaticly?

Here is my example: I have an index.jsx which I render server side and send to the client

var React = require("react");
var DefaultLayout = require("./layout/default");

var LikeButton = React.createClass({
  getInitialState: function() {
    return {liked: false}; 
  }, 
  handleClick: function(event) {
    this.setState({liked: !this.state.liked});
  }, 
  render: function() {
    var text = this.state.liked ? 'like' : 'haven\'t liked';
    return (
      <p onClick={this.handleClick}>
        You {text} this. Click to toggle.
      </p>
    );
  } 
});

var IndexComponent = React.createClass({
   render: function(){
       return (
           <DefaultLayout title={this.props.name}>
                <div>
                        <h1>React Test</h1>
                </div>

                <div id="testButton">
                    <LikeButton/>
                </div> 

                <script type="text/babel" src="/js/react.js"></script>
           </DefaultLayout>
       )
   }   
});

But the "Like Button" does not have any interaction. To make it do something on click I have to add this code client side.

var LikeButton = React.createClass({
  getInitialState: function() {
    return {liked: false};
  },
  handleClick: function(event) {
    this.setState({liked: !this.state.liked});
  },
  render: function() {
    var text = this.state.liked ? 'like' : 'haven\'t liked';
    return (
      <p onClick={this.handleClick}>
        You {text} this. Click to toggle.
      </p>
    );
  }
});

ReactDOM.render(
  <LikeButton />,
  document.getElementById('testButton')
);

I only started out with react.js and maybe I am just missing some major concept here. But why does react.js not just create the code (which I now have to add manually to the client) when rendering the page server side? Like this, I have redundant code and it feels like this will be a mess in larger applications. At least react.js is smart enough to not draw two LikeButtons but to "bind" the one created server side to the client side component.