It’s not something I consciously decided to do, but just judging from my instincts to say years; from 2001-2019 I’ll say two thousand n’one, two thousand n’two etc; “n” being the barest whiff of the word “and”. From 2020 onwards I revert to twenty twenty, twenty twenty-one etc.
2000 — two thousand
2001-2009 – twenty aught X *
2010-2099 – twenty XX
* Except in the movie/book title, Two Thousand One: A Space Oddity Idiocy Odyssey
I just skip the “n” altogeter. It’s not like we said nineteen"n"-ninety-four. I also use “twenty-oh-two” for the oughts. I am sure that is not grammatically correct.
2000-2009: Two-thousand, two-thousand one, two- thousand two…
2010 onward: Twenty ten, twenty eleven, twenty twelve…
That’s how I do it as well, and everyone else I know for that matter, although hearing the full “Two thousand fourteen” is very common as well for any year.
I just haven’t heard the “Two thousand and twenty one” or “Two thousand aught five” (except when someone’s deliberately trying to sound like Abe Simpson) in the wild, so to speak.
2000-2009: “Two thousand three”, or the like.
2010-2019: Either “Two thousand twelve”, or “twenty twelve”.
2020 onward: Mostly “twenty twenty three”, but I think I might be transitioning to “twenty three” without the leading “twenty”.
Exception: If I’m referring to a person by their year of graduation (common in my line of work), I shifted over to just the last two digits sometime in the teens. Someone who graduated in 2019 would be “Class of nineteen”, but someone who graduated in 2011 wouldn’t be “class of eleven”. I’m not sure exactly when I switched.
2001 - 2019: two thousand one, &c.
2020 - on: twenty twenty, &c.
This.
Two thousand, twenty oh three, twenty ten, twenty twenty three.
Like that.
Seventeen oh one, Eighteen oh one, Nineteen oh one, Twenty aught one (“twenty aught” sounds better than “twenty oh”)…
I arbitrarily switch between “two thousand-[number]” and “twenty-[number]” (or “twenty-oh-[number]” between '01 and '09).
I use both “twenty-oh” and “two thousand” constructions. I will say “two thousand one” to “two thousand nine”, but typically it’s “oh-one” to “oh-nine,” like “I moved back to the States in oh-three.” But I do sometimes go to the “twenty-oh” constructions, especially in the 2010s onward. I don’t do this “aught” business nor have I actually heard it in the wild. But I’m American, so it’s perhaps more common in other flavors of English?
If you’re actually saying “two thousand” you have to acknowledge the “and”. Failure to do so leads to linguistic madness 
My pet peeve is people who add “and”, eg, “two thousand and three,” or “two thousand and seventeen.” I’m pretty sure that this stems from a “rule” I was taught in elementary school that one does not insert “and” in numbers this way, which probably had all the validity of all those other so-called rules, like not splitting infinitives or ending sentences with prepositions.
But, right or wrong, it still bothers me every time I hear it.
I have heard at least one person refer to “twenty three”, “twenty six” — no pleonastic words
Back during 2001-2009 I usually used “two thousand one” through “two thousand nine”. But occasionally “twenty oh one” through “twenty oh nine”.
From 2010 onwards I’ve said “twenty ten”, “twenty eleven”, and now “twenty twenty three”. I also find I now refer to that first decade retrospectively more as “twenty oh one” through “twenty oh nine” than I did when living through those years.
The archaic “aught” / “ought” usage might have been tolerable a century ago about the early 190x years when nearly everyone only had a grade school education. At best. It has no place in the 21st Century unless one is a yokel or wants to sound like one. Ditto inserting an “and” or “'n” into a number. Just don’t. Please.
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Having been born near mid-century last time I think I lived through too much of the 19xxs and probably won’t live long enough into the 20xxs to be able to say things like “Back in oh eight we elected President Obama” without it sounding forced or confusing.
Somebody born after maybe 1985 could say “Back in oh eight …” unironically / unhumorously when later in life themselves. Whether that usage will ever become popular with that cohort is a different question. But IMO it could.
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An interesting side effect of the Y2K situation was that in the 19xx years I often just used the 2-digit year. e.g. In 1987 I might have said “I graduated high school in seventy six”.
Conversely, since the year 2000 and subsequent, I never refer to a year without the leading “twenty”. I’d never say “I moved to Florida in fourteen”. I’ll always say “I moved to Florida in twenty fourteen”. They always get written with all 4 digits too.
I’d be pushing age 150 when 2100 rolls around, but by cracky, I’m ready for it! :wrinkly old fart grin: :waving cane:
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Oops. Clearly I drop the “oh” in “twenty-oh” constructions from 2010 onward. 
Here in England, I and most people I hear (on the radio etc) say Two thousand and five etc, up to Two thousand and nine, although a few people leave out the “and”. After that, it’s twenty all the way. Twenty-ten, twenty-twenty-three etc.
By common usage, the first decade has become “The Noughties.”
Same for me.