I'm attempting to run a SWF in a browser and have it communicate to an AIR application that is bundled as an exe for windows.
In my AIR app (the receiver) I have the following
package
{
import flash.display.Sprite;
import flash.events.StatusEvent;
import flash.net.LocalConnection;
import flash.text.TextField;
/**
 * ...
 * @author
 */
public class Main extends Sprite
{
    private var _tf:TextField;
    public function Main()
    {
        var localConnection:LocalConnection = new LocalConnection();
        localConnection.addEventListener(StatusEvent.STATUS, handleConnectionStatus);
        localConnection.allowDomain("*");
        localConnection.client = this;
        localConnection.connect("_myConnection");
        _tf = new TextField();
        _tf.appendText("Creating local connection\n" + localConnection.domain + "\n");
        _tf.width = stage.stageWidth;
        _tf.height = stage.stageHeight;
        addChild(_tf);
    }
    private function handleConnectionStatus(e:StatusEvent):void
    {
        _tf.appendText("handleConnectionStatus\n");
        _tf.appendText(e.code + "\n");
        _tf.appendText(e.level + "\n");
    }
    public function testMethod():void
    {
        _tf.appendText("You wrote \n");
    }
}
}
And in the sender, the SWF on the web I have
package
{
import flash.display.Sprite;
import flash.events.AsyncErrorEvent;
import flash.events.Event;
import flash.events.SecurityErrorEvent;
import flash.events.StatusEvent;
import flash.net.LocalConnection;
import flash.text.TextField;
/**
 * ...
 * @author
 */
public class Main extends Sprite
{
    private var _tf:TextField;
    public function Main()
    {
        if (stage) init();
        else addEventListener(Event.ADDED_TO_STAGE, init);
    }
    private function init(e:Event = null):void
    {
        removeEventListener(Event.ADDED_TO_STAGE, init);
        // entry point
        _tf = new TextField();
        _tf.width = stage.stageWidth;
        _tf.height = stage.stageHeight;
        addChild(_tf);
        var localConnection:LocalConnection = new LocalConnection();
        localConnection.addEventListener(StatusEvent.STATUS, handleConnectionStatus);
        localConnection.addEventListener(AsyncErrorEvent.ASYNC_ERROR, handleAsyncError);
        localConnection.addEventListener(SecurityErrorEvent.SECURITY_ERROR, handleSecurityError);
        _tf.appendText("Creating local connection\n" + localConnection.domain + "\n");
        localConnection.send("_myConnection", "testMethod", "Hello world!");
    }
    private function handleSecurityError(e:SecurityErrorEvent):void
    {
        _tf.appendText("handleSecurtityError\n");
        _tf.appendText(e.text + "\n");
    }
    private function handleAsyncError(e:AsyncErrorEvent):void
    {
        _tf.appendText("handleAsyncError\n");
        _tf.appendText(e.error.message + "\n");
    }
    private function handleConnectionStatus(e:StatusEvent):void
    {
        _tf.appendText("handleConnectionStatus\n");
        _tf.appendText(e.code + "\n");
        _tf.appendText(e.level + "\n");
    }
}
}
I run the AIR app and then I run the SWF, but the SWF just prints out "error" for the event's level property. Everything I'm reading online says this should work. I try to run the SWF on localhost and from an actual website and it acts the same either way.