Since the Screen Sharing feature seems to be the only one unable to connect, the best way to diagnose its behavior is by using the application directly.
Please, launch screen sharing located in:
/System/Library/CoreServices/ScreenSharing.app
The application has no visible "UI" so don't expect a fancy Cocoa interface. Instead, once you have loaded it, all you'll see is this "poor" icon in the Dock:
Click on it and then go to Preferences in the Mac menu bar. You should be presented with:
Notice the second option to only encrypt passwords and keystrokes. That's fine, but as you can see, there's no option to skip encryption altogether. Some VNC versions have incompatibilities with encryption, but this would be rather shocking since you're talking about OS X vs OS X.
In any case, once you have the settings like that, try connecting, again, in the menu bar: Connection -> New (or ⌘cmd + N).
A simple dialog appears asking you to enter the Host's name. Following your screenshots, try: 192.168.1.99 and hit enter.
We'll see what happens from here. Also, have the Console.app open on both computers, to see if there are any System messages going on (or anything related).
ARD into your "work account" and then do the following on the remote computer to open a Screen Sharing window to your "personal account" on the same remote computer.
Create an SSH tunnel. It works for me using 10.9.
Enable Remote Login (SSH) in System Preferences > Sharing, and in Terminal run:
ssh -NL 5901:localhost:5900 localhost
Finally, use Screen Sharing to connect to localhost:5901 (instead of the default port 5900). Instead of port 5901 you can use a much higher port that isn't in use by anything else.
The Screen Sharing application is located here: /System/Library/CoreServices/Screen Sharing.app
Best Answer
There are two parts to remote sharing your Mac computer: software on the computer and configuration of your network.
Software on your Mac
You can use the built-in VNC server on your Mac; there is no need for third party software.
Next, test you can access and control your Mac using a VNC client within your home network. This can be done from a Mac, Windows, or mobile device using a VNC client.
Configure your network
To access your Mac from outside of your home network, you need to open a port for VNC:
5900TCPIf this port is not open and passing traffic to your Mac, then you will not be able to connect from outside your network.
How you configure your modem/router differs between the make and model of each device. Some routers will have built-in assistants to help set this up.