Twitter etc should really already have standardised on a way to store the ALT text in an image so it only ever needs to entered once.
Even GIF has support for text in a couple of different forms.
( Although I learned something new about PNG's support for GIF's text support here: http://www.libpng.org/pub/png/book/chapter11.html#png.ch11.d... )
( I've spent far too long reading the GIF spec in relation to my project that adds audio to GIFs: http://audiogif.rancidbacon.com :) )