Here's a recap of my latest Skype experience on my S4. If you recall, Google Play version of Skype didn't work for me. I uninstalled it and tried the hacked version that SebaKL linked to. I tried to Skype w/ a family member on a different network (I'm VZW, she's on Sprint) and it didn't work (as in just audio, but no video). It works for her on her HTC One when she Skypes w/ her phone to my PC.
Then I tried to Skype w/ another family member, who also has an S4 on VZW like me. I first asked him to use his PC, which I know works. It took a few call attempts, but it finally worked, we were both able to see each other. Then I asked him to use his Skype app (he also has the Google Play version). After a few more call attempts, that also worked! (FWIW, we both used our respective Wi-Fi networks, not VZW's 4G LTE.) The audio was a little buggy. On some calls, the speakerphone worked. On other calls, only the earpiece volume worked.
So, in a nutshell, we managed to get video calls on the Skype app to work, but it just requires some patience (i.e., some call attempts). I'll stay w/ the hacked version rather than go back to the Google Play version. Hopefully this will help out future S4 Skype app users.