Try holding your finger on the outside of the message and a menu should pop up. See if you see the option "copy to sd card" & tap that. If you don't see that option you can use the app called save mms to save pics
I have had this same issue and when I press the screen to bring up that option it flashes on the screen the goes away. So I am not able to save the photos to my phone either.
Another option you have when looking at the picture is to select the message. Once you have selected the message by either a long press or pressing the menu button. It will then give you the options of what is available. You should then be able to save the message to your media card. Hope this helps.