The only references I’ve heard to this decade in the media are “at the turn of the century” on some news interview shows or “this decade”. Since there’s no true official name, only what people use, I guess we’re stuck with those.
I’ve been calling it “this crappy decade”. When it’s over, I’ll call it “that crappy decade”. There will be no confusion.
I use this, essentially, but I use the “oh-oh” pronunciation that doesn’t come across the same way in print.
As in, “Yeah, George W. Bush was president for most of the decade we call the oh-ohs. It was a great time to be a comic.”
They did a flashback show for a decade that hasn’t even ended yet?! 
Well once again the parody of the Onion proves to be merely a harbinger of reality to come. They predicted this eventuality over 10 years ago, in an article entitled U.S. Dept. Of Retro Warns: ‘We May Be Running Out Of Past’.
As to the question in the OP, the most common expression I’ve heard/read is the rather uncomfortable phrase “this decade”, which has not much longer to go for it, followed by “the 2000s” (“two thousands”).
I call it the 0s. Pronounced “the ohs”. You say oh-2, oh-3, oh-4 don’t you? So that’s the simplest and most obvious way of referring to the decade.
While playing a trivia game with a friend where you try to guess the year (of a song, movie, etc). We try to quickly zero in on the decade (or half), then guess at the year.
When a certain question came up, he yelled “Late Nineties”. “No, No” I quickly replied, “Early Naughts”. Since he knew what I meant, I guess that term is close enough (or “oughts” if that’s how you roll). Naughties and oughties is just stupid.
When I hear " two thousands", or “nineteen hundreds” I think of the whole century, not the first decade (in fact, that’s what I thought the OP meant until I read the thread).