It doesn't directly have anything to do with Sleep. Both of the work Macs that I have problems with are set to Never sleep.
However, I did try something today that seemed to affect it immediately. While showing something to someone in the Sharing preferences, it occurred to me that just because the File and Screen Sharing options were checked that didn't necessarily mean that the daemons were running (or running properly). In fact the Error message I get if I try to vnc://<myhost>> is that I should check to make screen sharing is turned on (which, of course, I have done, by looking to make sure the checkbox is checked). So I unchecked them and then rechecked them on both systems, after which, they showed up in each other's Finder windows almost immediately. indicating that (re?)starting the daemons fixed the issue. As of now, they are happily connecting to each other. I'll see if I can connect later from home, as I do put that Mac to sleep when I am not there.
One thing about this is that I mentioned above that it starting working for me after upgrading to Safari 8.0.2. It's possible that something that happened during the Safari 8.0.2 upgrade (either as part of the upgrade or coincidentally) triggered these daemons to (re?)start. Thus making it appear that the update weirdly somehow affected the issue. Does anyone happen to know the process names for these daemons?