Just travel to egypt, rummage around the pyramids and find a Millennium Eye, thats how I did it, and I'm now able to see their cards.
But no, jokes apart, you should be able to do it by checking incoming images to your cache, as the game makes you load their cards image as soon as it reaches their hand, so that you wont get this black card with a loading in the middle that you see sometimes. so if you can inspect the incoming data/images you should be able to see the image of their card before they use it.
Also, incase any programmer reads this, would it be possible to alter ID's, like changing which card you have on hand? doing this you could win any match by simply going for exodia in your first draw :P obvious, but would never fail.
Incase anyone wonders, the game is a simple Flash game, and im not sure that the data is server based, might be client based.
Also, the game doesnt have any enforced rules, it leaves it up to they players to enforce the rules, meaning, the game cant say "No you cant do this" only the other player can if he can see what you did.
EDIT: after some playing around i was quickly able to find that my theory worked, i was playing against a friend, and he drew exodia from his deck, and guess what popped up in my cache folder? this
https://www.duelingnetwork.com:8080/D...cards/1082.scf