Hi,
In Yosemite it is very clear that there are two forms of Screen Sharing (and Video and Audio Chats).
As you are probably aware there was iChat before there was Messages.
AIM and Jabber accounts in iChat could video and Audio Only chat to other Mac/iChat users and also do Screen Sharing by adding a VNC connection to an Audio Chat.
(Jabber accounts included using Google IDs and Facebook Chat IDs).
This continued into Messages.
Soon after that the iMessages Contacts could then link/start up FaceTime to Audio Only or Video chat.
As this was to iPhones as well as Mac it made some sense - this invoking of another app.
In Yosemite Screen Sharing was added to this.
It invokes the Screen Sharing App after you have Enabled permission for it to be called upon in System Preferences > Sharing.
You then used the Details item top right of a Chat when having an iMessages conversation.
(If you do this in a AIM or Jabber Chat it tries to use your iMessages ID to connect and will not work)
In El Capitan the Video Menu no longer has the AIM/Jabber Screen Sharing option listed.
When you are in a Jabber or AIM chat then the Buddies Menu makes it clear you can Video or Audio Only chat.
The issue with that is that there is a problem with the SNATMAP server that Apple run that connects these type of Chat.
Computers on a LAN work (So Bonjour works but it can be AIM to AIM if you have different AIM valid Screen Names on two computers)
What does not work for many people is Internet Chats using AIM and Jabber IDs
The iMessages Screen Share has worked since it was introduced.
I have not heard of an issue with it - if the Screen Sharing is Enabled in System Preferences.
What I am saying is that there can be a clear difference between Buddies (AIM and Jabber) and Contacts (iMessages) with regard to which way they Screen Share.
I also find Buddy Lists show differences even when listing Buddies.
I have 4 Buddies based on other AIM valid accounts I uses in Messages.
Only one says it can Screen Share according to the icons


10:08 pm Saturday; November 7, 2015
iMac 2.5Ghz i5 2011 (Mavericks 10.9)
G4/1GhzDual MDD (Leopard 10.5.8)
MacBookPro 2Gb (Snow Leopard 10.6.8)
Mac OS X (10.6.8),
Couple of iPhones and an iPad