Hi,
Can I check whether you are using iMessages as a Starting point or is it a Google (Jabber) account in the Messages App ?
In theory the Messages app should be able to start two forms of Screen Sharing.
The Method that was present in iChat and Messages upto Mavericks as the only method means working form either a visible Buddy list or the Buddies Menu - having Enabled Screen Sharing in the Video Menu (along with Audio and Video).
At Yosemite an iMessages method is added.
This means starting an iMessages text chat then using the Details option top right of the Main Chat window.
The original advice was that System Preferences > Sharing > Screen Sharing had to be Enabled and allow Anyone.
In El Capitan the Buddies Menu does not have "Start Screen Sharing" or any of the previous wording. The Video Menu does not include an option to Enable Screen Sharing. (this is only for AIM and Jabber accounts and Google is a Jabber account).
If you start an AIM or Jabber text chat then use the Details Option it will try to start an iMessages Screen Sharing.
(i.e. it is crossing over functions).
NOTE.
The iMessage method is like the FaceTime Video chat in that it is invoking an Outside App to do the task.
So...
You can in theory Screen Share Jabber to Jabber (Mac to Mac) AIM to AIM or use the Screen Sharing app via the iMessages text chat.
The next problem comes with the older iChat Method.
These don't appear to be working at the moment as all AIM and Jabber attempts go through an Apple server called SNATMAP.
This does not appear to be working properly.
It will let Bonjour Video Chats work and if you have two IDs such as two separate AIM Screen Names on two computers on your LAN then those will connect.
There are issues connecting over the Internet with Buddies.
This has been brought to the attention of the Hosts here and they state they have passed it on.

8:46 pm Tuesday; November 24, 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