The latest update to OS X Mountain Lion Developer Preview 3 seems to have partially enabled iOS-style automatic app downloads in the App Store. Like on the iPhone and iPad, when you buy and install an app on one of your Macs, all of your other Macs logged into the same App Store account will automatically install the app too.
Unfortunately, the feature does not seem to be working completely. While the App Store will still offer to enable automatic downloads (as seen above), it does not actually install anything when you purchase apps from another computer. However, it does apparently show the app as being installed when you view it in the App Store. There is a good chance we will see at least one more developer preview, or at least a GM, in the coming weeks that should fully enable the feature.