They’re not particularly well drawn, and unfortunately will override native Emoji on operating systems that support them, such as Safari on Mac OS X.
As with Apple’s Emoji set, this artwork is most likely their copyright and not to be used outside of Twitter properties. However, I have no idea what the source of the artwork is. I might assume that they created the artwork themselves, except that oddly the image they’ve used for “Bird” is red rather than blue. Perhaps that avoids confusion with the Twitter logo, or perhaps they just missed the opportunity. Who knows? or as you’re more likely thinking… who cares?
Twitter’s copyright interests include “other distinctive brand features”. I’m no lawyer, but as most emoji icons are simply redrawn or traced versions of various other sets, I’d hardly call Twitter’s Emoji artwork “distinctive” in the context of their brand. Anyway, when they email me with a takedown demand, I shall tearfully comply.
If Twitter did develop the icons themselves, it’s a pity they didn’t contribute their effort towards an open source set of Emoji – particularly as the Kickstarter project “Phantom Emoji” ran out of money before the set was anything close to complete.