If you are using FB connect module and its Invite friends submodule or making your own invite FB friends submodule (which is easy). You could probably get into a problem of having no notifications sent to user when you do click Invite in FB dialog box.
There are 2 possible reasons for that. One your app is missing the canvas configuration
Just add it from the your App panel in Facebook Developers. Currently it is in settings tab (4/2014) on link https://developers.facebook.com/apps/383569161782956/settings/
where you need to click "+Add platform" and then choose "App on Facebook" and then add URL of your site there and save.
The second one - make sure your app is NOT in SandBox Mode or currently called Dev mode. If it is it probably won't send any notifications. 
and for making a invite form all you need to is add this code to some button.
<script src="http://connect.facebook.net/en_US/all.js"></script>
<script>
FB.init({
appId:"399804580159414",
cookie:true,
status:true,
xfbml:true
});
function FacebookInviteFriends()
{
            var request = {
                message: "my message",
                method: "apprequests"
            };
            FB.ui(request, function (data) {
                if (data.request_ids.length > 0) {
                    alert("Facebook invite successful.");
                }
            });
        
}
</script>
//HTML Code
<div id="fb-root"></div>
<a href="#" onclick="FacebookInviteFriends();"> 
Facebook Invite Friends Link
</a>'