Wrap sibling groups using jQuery.

jquery.js

$(document).ready(function() {

  $(':not(.sibling-element) + .sibling-element, * > .sibling-element:first-of-type').
     each(function() {
       $(this).
           nextUntil(':not(.sibling-element)').
           addBack().
           wrapAll('<div class="sibling-wrapper" />');
     });
});

result.html

<div class="entry-content">

	<div class="sibling-wrapper">
		<div class="sibling-element">Sibling</div>
		<div class="sibling-element">Sibling</div>
	</div>

	<p>A random separator</p>

	<div class="sibling-wrapper">
		<div class="sibling-element">Sibling</div>
		<div class="sibling-element">Sibling</div>
		<div class="sibling-element">Sibling</div>
	</div>

	<p>Another random separator</p>

	<div class="sibling-wrapper">
		<div class="sibling-element">Sibling</div>
		<div class="sibling-element">Sibling</div>
		<div class="sibling-element">Sibling</div>
		<div class="sibling-element">Sibling</div>
	</div>

</div>

shortcode-output.html

<div class="entry-content">

	<div class="sibling-element">Sibling</div>
	<div class="sibling-element">Sibling</div>

	<p>A random separator</p>

	<div class="sibling-element">Sibling</div>
	<div class="sibling-element">Sibling</div>
	<div class="sibling-element">Sibling</div>

	<p>Another random separator</p>

	<div class="sibling-element">Sibling</div>
	<div class="sibling-element">Sibling</div>
	<div class="sibling-element">Sibling</div>
	<div class="sibling-element">Sibling</div>

</div>