Seems to me that the answer is “both,” because both fit standard well-established patterns of English speech. On top of that, both are clear in context.
The One Right Way theory leads some people to insist that it must be just one, usually by analogy to other usage. Some, for instance, argue that it must be “twenty ten” by analogy to “nineteen sixty-eight.” But why not “two thousand ten,” by analogy to “two thousand”? (Never mind the common “two thousand nine” locution, since some of the “twenty ten”-only advocates claim that too is somehow wrong.)
These sorts of analogies work only to the extent that they help one remember or recognize actual usage patterns. Here, as best I can tell, both “twenty ten” and “two thousand ten” are in common use. You can choose whichever you wish for yourself, but there is no reason to treat one as right and the other as wrong. Prof. Arnold Zwicky has more on this.
Incidentally, this is the way things appear to be in English; common usage in other languages may be different. For instance, the way I learned Russian — and I should acknowledge that I doubtless wasn’t exposed to the wide range of Russian usage, especially current Russian usage — one might say the Russian equivalent of “one thousand nine hundred seventy fifth,” or “seventy fifth” (the word “year” is implied, though it could also be explicitly included) but never “nineteen seventy fifth” or “nineteen hundred seventy fifth.” Saying “Twenty tenth” would thus be incorrect in Russian, simply because it would be a departure from all of the standard usages. (Likewise, saying “ten hundred” would be incorrect in English, though my six-year-old keeps using it, since it’s perfectly logical and analogous to nine hundred, fifteen hundred, and the like. It is a departure from all of the standard adult usages, except when one is trying to emphasize some mathematical fact or analogy, for instance when teaching a child that ten hundred equals a thousand.)
Yet all this further illustrates that the sensible test of correctness is usage, which varies from language to language, and not universal logic. Nor can the test be some hoped-for consistency within the language, as anyone who knows the Russian word for 40 (“sorok,” completely inconsistent with the pattern for other multiples of ten) or the English words for 11 and 12 (completely inconsistent with the pattern for 13 to 19) can tell you. A language consists of what its speakers say, not of what would-be logicians prefer.