I find that I use both "twenty sixteen" and "two thousand sixteen", with a slight preference for "two thousand sixteen".
I think if I'm thinking about the present, I say two thousand sixteen. If I am thinking about history - for example comparing the 2016 election or the 2016 baseball season to past elections or past baseball seasons - then I say twenty sixteen. I guess this is because if I'm also thinking about 1992 or 1824, my mind is thinking "eighteen, nineteen, twenty..."