Usually long pressing the image gives you the option to save the image to the phone.
I'm most definitely NOT a VZW employee. If a post answered your question, please mark it as the answer.
Ann is right...just press an hold and you should see an option to Copy, Save, or More (which allows you to forward to someone else). If you actually tap on the image (to open the picture) rather than holding directly in the text thread, then you won't get the option to Save...it'll only show Copy.