Yes, you can take screenshots of every game at anytime, and copy them to an USB drive, or upload them via Twitter and Facebook (unless the game forbids you, to stop you from posting heavy spoilers, for example). Hell it even save videos of your gameplay automatically, and it's very easy to deal with this stuff, copy, upload it, edit it, etc.
And what I mean "paying for it" it's because some games on the Xbox 360 have a photomode, in which you can take pictures of the game (Forza, for example). But the problem is: you can't upload them if you don't pay for a Xbox Live subscription. And you can't even copy them to an USB drive to see those pics on you PC. F*ck, you can't even have a Netflix subscription on the Xbox without having to pay for a Xbox Live Gold account. It really sux having basically everything behind a paywall.