With an upcoming feature that Microsoft has not yet announced, but hasn't denied working on, you might be able to! Ok, so you can see that live already now, with Xbox Live Vision, but it isn't as cool as saving it and sharing it with your friends, right? Wait, what?

Recently a game developer let slip to Kotaku (inadvertently) that Microsoft was working on universal video capture...
Presumably this would be like the video capture included in Halo 3; you would be able to take videos and screenshots, edit them, upload them, and share them. During an interview the game developer (whose name is being kept anonymous to prevent backlash from Microsoft) was discussing video capture, and had this to say: "We'd love to have video capture in our game, but why work on something that the platform holder is already developing."
Microsoft Xbox Live digital games director Scott Austin had this to say about the possibility: "We are always thinking about ways that Live at the platform level can make games better. I'm not going to comment on video capture, but you can assume we are thinking about ways to make things more interactive." Imagine - me going end-to-end in NHL 10, flying through the neutral zone, toe-dragging past a defenseman, and firing the puck from the top of the circle past a hapless Brodeur...then being able to immortalize it as a captured replay for all to see.

So yeah, it would be more like me getting hit in the ass with the puck and then having the goalie nudge the puck into his own net trying to stop it, but hey, scoring is a momentous occasion for me, it is worth saving regardless! Being able to save footage from games you play would be amazing. Amazing goal/frag/trick? Display it for your friends. Find a cheater? Report him, with evidence. Find a bug?
Share it with the developers, maybe...Beyond that, when you start talking about "universal video capture" the question of whether or not this may be available outside of games comes to mind. There seems to be a wealth of untapped potential there as well. So what does all of this have to do with naked video sharing? Nothing really, other than the fact that even if the video capture is only available inside of games, those that allow you to see live streams from players' video cameras, like Magic: The Gathering and Uno, would presumably be recorded for everyone to see. Now you'll be able to show proof that BigBubba was whipping how his pea-shooter in front of the kids. So, be prepared to have all of your most embarrassing (digital) moments saved and shared with the world, hopefully soon.