I tried that before making this post. I downloaded and re-installed 10.8.2 because I started with 10.8.1. I wanted the combo update and an updated driver from Epson on the App Store as an update. That gave me a fresh OS and driver, but it didn't solve the problem. Epson's tech support was not helpful (IMHO).
My best guess is that another update is needed. I can disconnect the USB cord from the other Mac and plug it into my machine and things work fine, but I can't get it to successfully do it through sharing. I may not know enough about sharing to do it right, but my settings were good enough to work perfectly when both Macs had Snow Leopard. The other Mac is staying with Snow Leopard for the sake of Rosetta.
Yesterday, I was desperate for a scan and the other Mac was very busy, so I restarted with Snow Leopard on my machine and made the scan. I then sent myself an email (with the scanned image attached) from Snow Leopard and quickly shut down. I then restarted from Mountain Lion and opened Mail. Voila! there was the message with its attached image. Frankly, that is NOT an effective use of resources, but it did solve an immediate issue.
