Firstly, in order to integrate into Notification Centre, an app needs to have been purchased from the App Store. Only App Store apps can use it (same with most iCloud features). So your search needs to start there.
2 very popular RSS readers are Reeder and NewsNetWire. Neither have been update to use Notification Centre yet so far as I know, but for sure some of them will be.
It's likely that if you can find an RSS reader that used to use Growl as it's notification method, then it will be modified in time to use Notification Centre instead, but there are so many RSS readers (particularly on iOS, perhaps less so on the Mac App Store) that I can't name a particular app right now.
I quick search for "rss notification" yields no results, but "rss growl" gives 3 hits (Note that for those who are not aware, Growl is in effect a third party tool that has done what Notification Centre will do for years already, and loads of apps already support it):
http://itunes.apple.com/gb/app/cappuccino/id492594411?mt=12
http://itunes.apple.com/gb/app/monotony/id533978766?mt=12
http://itunes.apple.com/gb/app/feeds/id458374415?mt=12
The fact that these 3 already support some form of notifications may be indicative they they will be amogst the first to support OS X 10.8 Notification Centre.
However there is another way:
For those apps (not just RSS) that use Growl, there is talk that Growl will morph into a sort of bridge between the Notification Centre and non MAS apps, because it is on the MAS itself, and as such can install notifications into the tray, and it can also poll other apps for Growl notifications and simply pass them on without the originating app needing to know about Notification Centre formally. See this blog post for info.
It's not happened yet tho, although there is already an app to bridge between Growl and Notification Centre called HISS (See here http://9to5mac.com/2012/07/25/put-growl-in-your-notification-center-with-hiss/) which will punt any grown notification into NC instead, although it's very much Beta, and frankly will be superceded as soon as Growl get their own act together, but at least it is available now.
As such you may be able to roll your own temporary bodge using a combination of an RSS reader as suggested above that supports Growl, installing and using Growl, installing Hiss to pass Growl notifications into the Notification Centre...
I'm a Mac OS X developer and I just finished adding Notification Center support to my app. For what it's worth, here's what I found out:
- By default, notifications (the alerts) are NOT shown if the application is already frontmost (this explains why alerts refuse to show up sometimes, and you only see the notification in Notification Center). This is up to given applications to overwrite and it is not very straightforward, so do expect most applications to not show notifications when they're active. In my opinion, this is very annoying.
- The application, after sending a notification still has a lot of control over that given notification. It can remove it from your screen at any time it wants, regardless of the settings you have. Basically, if Safari, for some reason was set to remove notifications after X seconds, it will do that, your settings do not matter.
I have no idea if this is a bug in Safari or Notification Center, or both. However, I am fairly confident that there is nothing you can do about the second issue (there might be a hidden defaults setting for the first one, but I haven't heard of it). I did check the WebKit API Specification, to make sure it's not something that the individual websites are doing, and you can rule out websites from your suspect's list, because they do not have any control over the notifications.
I guess we'll have to wait and see if Apple will address this in its next bug fix release (10.9). However, by the looks of it, not showing notifications when the app is frontmost is a "feature", so don't get your hopes up.
Best Answer
When you go to Gmail in Safari 6, you should see this at the top:
That should enable alerts in Notification Center. You could also set up your Gmail in Mail.app.
On Apple's website for Mountain Lion, it says:
So it may be a few weeks still before you can use Facebook with Notification Center.