I should add a clarification and tip.
I want to make sure people in this thread are not confusing "networked" scanners with "shared" scanners. The Epson Workforce 610 has full network support so you should be able to connect to it directly. This is just to point out that "Share this scanner on the network" does not need to be selected to solve this thread, and in fact that option is completely irrelevant to the problem in this thread. The "Share this scanner..." feature is for scanners that cannot be connected directly over the network; the feature lets a Mac do the work of sharing the device over the network. But the Workforce 610 does not need this option because it can share itself. (Image Capture makes this a little confusing because it lists networked scanners under the "Shared" heading.)
The tip is that yesterday I could not reach my Workforce 610 over the network even though I had just posted that I could. 🙂 This is something that happens occasionally to me that is not related to the Workforce or Mavericks specifically, but once in a while I find that I need to restart my router and then it all works again. So I wanted to make sure that, as part of the troubleshooting, the router has been restarted at some point.
Because once I restarted my router, the Workforce 610 resumed appearing as a scanner across the network in Image Capture, using Mavericks.