Experiencing the same issues as everyone else. To date, AT&T has been clueless and it certainly doesn't seem like anyone has any specific information on this issue. Like you, shortly after I received the Android 4.4.4 OTA update, I started noticing that in group MMS messages, I was missing content. Also, I started getting the "Anonymous" messages and the "Download" messages that fail to do anything. I contacted AT&T and have since spoken with no fewer than 5 different technical representatives. The last of which, sent me a warranty replacement as a last resort after checking APN settings, deleting all messages, factory resetting my phone, etc. etc.
I received the replacement yesterday and got a brand new SIM card for it. After updating all of the apps and receiving the OTA update to Android 4.4.4, I'm still having the same issues. According to a store associate, AT&T is aware of the issue and waiting on a fix from HTC; however, in speaking with someone in technical support, there were no open "known issues" accounting for this MMS problem. That said, if you haven't done so already, please call AT&T and let them know about the problem to generate some more awareness.
This is so incredibly frustrating being that the issue has persisted for more than a month, AT&T doesn't seem to know what's going on, and I even communicated with HTC who also do not appear to be aware of the problem. Being that the last resort (warranty replacement) has failed to fix the issue, I'll be calling customer support once again to see if they can will send me a replacement phone of a different variety. I like the HTC One M8, but frankly, I'm not willing to wait the months it's likely going to take to sort this issue out and get a software update to fix it.