I think the reason this may be happening to some people is that they are using the default configuration and don't have the Curl functions in PHP.
The default config.php uses Curl and not all PHP installations include PHP-Curl. If you have the ability to check your web server logs you may see something like:
PHP Fatal error: Call to undefined function curl_init()
If this is the case you need to either change the line in config.php from:
$addon_conf['ArmorySync']['UseCurl'] = true;
to:
$addon_conf['ArmorySync']['UseCurl'] = false;
Or figure out how to get Curl built into your PHP implementation.
Also, I didn't have PHP-Curl and the FileSocket option wasn't working for me either. Still not sure why the FileSocket function was not working, but when I added PHP-Curl everything started working. I will try and do some testing with the FileSocket function.