The only way to achieve this is to use a database (flat or relational, e.g. a file or MySQL). Then use serverside code to show the content on each page.
Assuming you want to update content in one place and have it displayed in all areas that use the same content, you can use php include to do this.
For example, if this is the content you want on both div's on 2 different pages:
You can put that code into a .php file - something like test.php. And let's say you have home.php and services.php where you want the same content to be displayed, add this line of code in both the pages:
<?php include('test.php'); ?>
This will include the contents within test.php file in the section where you've inserted the <?php include...> code.
Note: Your files should be .php for the php includes to work.
Divs don't communicate with each other. They are nothing more than placeholders for content (headings, paragraphs, lists, links, etc...).
Depending on how you build your site, content inside divs can be static (plain HTML) or dynamic (generated by the server from commands in your code).