If $post_string is a string, then no. Changes done to it in the observer class would not be shown in this scope because strings aren't passed by reference to functions.

There are several solutions for that:

Quick but not recommended:

PHP does give you a workaround, by forcing the string to be passed by reference to the function by adding an & before the the variable name, like so:

Mage::dispatchEvent('my_api_delete_before', array('post_string'=>&$post_string, 'my_cart'=>$this));

But the reason this isn't recommended, as @dedmeet said in his comment:

using call-time pass-by-reference is not supported since PHP version 5.4.0

Directly from the PHP manual:

Note: There is no reference sign on a function call - only on function definitions. Function definitions alone are enough to correctly pass the argument by reference. As of PHP 5.3.0, you will get a warning saying that "call-time pass-by-reference" is deprecated when you use & in foo(&$a);. And as of PHP 5.4.0, call-time pass-by-reference was removed, so using it will raise a fatal error.

So here is how to do it in a way that is

Cleaner and recommended:

A better solution would be to create a Varien_Object because classes are always passed by reference. And any class that extends Varien_Object and the Varien_Object itself, gives you the ability to use the getters and setters you find around Mangento.

$obj = new Varien_Object();

// Set a value for data variable
$obj->setFoo($bar);

// The previous function is the same as the following:
// In fact, the previous function will call setData with 'foo' and $bar as its paramteres
$obj->setData('foo', $bar);

// To get the value of 'foo' you will need to call any of the following functions:
$obj->getFoo();

$obj->getData('foo');    

So to implement the Varien_object in the OP's example:

class Foo_Bar_Model_Communicator{

    public function someFunction(){
        /*
         * Assuming that $post_string is getting set here somehow
         * before the next line.
         */
        $eventData = new Varien_Data();

        $eventData->setData('post_string', $post_string);   
        $eventData->setData('cart', $this); 

        Mage::dispatchEvent('my_api_delete_before', array('event_data'=>$eventData));

        /* In this case, since we sent a Varien_Object as parameter,
         * whatever the Observer class does to the object, will be reflected
         * here as well.
         * We only need to make sure that we get the string back from the
         * Varien_Object instead of using $post_string.
         */
        $new_post_string = $eventData->getData('post_string');

        $result = $this->doApiCommunication($new_post_string);

        /*
         * We can do the same for the parameters for the next event, but it
         * wasn't part of the OP's question and it won't be needed since
         * there is no code following the dispatch event.
         *
         * However it would be a better idea to make the "before" & "after"
         * events have the same type of parameters which would be less 
         * confusing later on.
         */

        Mage::dispatchEvent('my_api_delete_after', array('post_string'=>$post_string, 'result'=>$result, 'product'=>$product, 'cart_item_id'=>$_item->cartItemId));
    }
}