When connecting to a computer using Apple's "Screen Sharing" feature, you may run into a problem where the system pauses and shows the spinning color wheel and does not connect. The problem happens immediately after authenticating the connection, and usually shows a black "Screen Sharing" window along with the color wheel.
This may happen only when using one computer, and only when using that computer to view the screen of another computer and not the reverse. This indicates the problem is with the computer's client software configuration, and not with its ability to serve screen sharing to other computers.
Since the problem happens immediately after authenticating, the problem is likely with how the computer is managing the authentication processes, instead of a problem with the screen-sharing program itself. As a result, the best way to tackle this is to try quitting and relaunching the authentication agent; however, there are some additional steps to take to prevent the problem from occurring.
Open Activity Monitor and locate the process called "NetAuthAgent." Once quit, this process should allow screen sharing to quit and relaunch to try again. This has resulted in proper behavior for some people.
Remove keychain entries for screen sharing
If you have stored passwords for your screen-sharing processes, try removing them using Keychain Access since faulty or duplicate keychain entries can sometimes cause the authentication process to hang or provide incorrect credentials. You might also consider running "Keychain First Aid" on your keychains using Keychain Access.
Remove screen sharing preferences
Try removing the file "com.apple.ScreenSharing.plist" from the /username/Library/Preferences/ folder, and then try the screen-sharing session again. A few people have been able to overcome Screen Sharing hangs by changing settings for the program, which indicates some fault in the preference file may be contributing to the problem. In addition to removing the preferences, try running general system maintenance routines such as fixing drive permissions, and booting into Safe Mode then rebooting.