Apple's Screen Sharing feature is a convenient way to access a remote Mac and use it as if you are sitting right in front of it, when coupled with the Full Screen features in Apple's latest versions of OS X, you can even make your current Mac appear seamlessly to be the remote system.
Screen sharing can be activated over the local network, or from a distance using Back To My Mac or similar dynamic DNS services, and while Screen Sharing has its uses, there may be times when you find that you cannot properly interact with your remote Mac. Even though the screen sharing session is properly established and clearly showing your remote system's current desktop, clicking or dragging, and accessing menus simply does not work.
This may be frustrating if you are expecting to quickly make a change on your system; however, it is likely occurring from a simply setting in the Screen Sharing program. While Screen Sharing offers full control modes by default, it also supports an "Observe Only" mode which when enabled will only allow you to view the remote system and not make changes. This mode can be useful if you would like to maintain the remote session but prevent interfering with someone who is sitting at the system.
To toggle between Full Control and Observe modes, you can either press Option-Command-X, or toggle this setting from the View menu. Once back in Full Control mode, you should be able to interact with the remote system again.
If you find all of your screen sharing sessions regularly go into Observe mode, then you may be able to tackle this problem by removing the screen sharing preferences file, which is called "com.apple.screensharing.plist" and is located in your account's preferences folder. To get to this folder, hold the Option key and choose Library from the Go menu in the Finder. Then open the Preferences folder, and locate and remove the file. When done, try establishing a new screen sharing session.