You can't share your computer screen when you are using Adobe Connect 8 SP1 and the Mac OS add-in for that release (9,4,79,0) on Mac OS.
When you choose to share your screen, you receive a dialog box asking you to first enable this feature by enabling "access for assistive devices."
Choose System Preferences > Universal Access > Enable Access For Assistive Devices to enable this option.
Adobe is addressing this issue in an update for the Adobe Connect Add-in, version 9_40_80_0. If you haven't downloaded the add-in, you will receive it automatically via lightning download when you open a Connect Meeting Room. If you have the older Add-in, it's necessary to manually download it via Connect. (For example, download it via the Connect Getting Started page under the Resources tab on first login.)