I have a GS3, but I am experiencing this identical issue on 4.3
The dilemma (coming from 4.1.2) is that without checking "Auto Network Switch" under advanced wireless settings, I could not send or receive any MMS. I was getting the download prompt for inbound mms which never downloaded, and the sending message for outbound. Once unchecked, MMS works as it should. And as is pointed out here, that is somehow because the phone chooses mobile data over wireless because the connectivity integrity check of wireless determines 'Network disables because Internet connection is slow'. Which is not the case at all. I can clearly see my signal strength is excellent and the bandwidth test confirms that I am getting a good deal of my 75/35 FiOS pipe.
This only leaves the following options:
1. Uncheck 'Auto Network Switch' (under wireless>advanced) which will reconnect device to Wireless AP, but prevent MMS in or out. (should be noted that on 4.1.2, there was no 'auto network switch' option. If you were on wireless and sent/received mms, it briefly connected to mobile data to handle message.)
2. Check 'Auto Network Switch'. This allows MMS to work as expected but only because your device remains on mobile data. Indefinitely. No matter how good any wireless connection actually is, the phone defaults to disabling Wi-Fi in favor of mobile data. I can confirm there is no issue with my network connection being slow. I have excellent signal to my AP and to the Internet.
Desktop Speedtest Results
Mobile Speedtest Results
So this issue appears to be related to more than one device... is this an overall Samsung 4.3 issue?