Do "Click to Tweet" and "Click to Post" still exist? Mountain Lion had "Click to Tweet" and "Click to Post" buttons in Notification Center if you had Twitter and Facebook configured in System Preferences>Internet Accounts. However, I can't find them. I'm using El Capitan. Did Apple remove this feature? I can't find anything on the Internet saying so.
You have to add the widget. Open Notification Center, choose Edit, then drag the Social widget.



(The Twitter widget in the screenshot is from the Twitter app and shows tweets. That's not what you want. You don't need Twitter installed for this — Social is a system widget.)

If Twitter/Facebook doesn't show in the Social widget, make sure the Share Menu option is enabled for the account in System Preferences → Internet Accounts.