I think Decomposed Words are
those for which, if you extract the letters that are used for Roman numerals, make a prime number
and Not Decomposed Words are accordingly
those which yield either a composite number or 1.
The given examples don't let us tell
how to classify words that contain no such letters.
Per GentlePurpleRain's suggestion in comments: the name is because
when you decompose a number as far as possible into factors, the components you end up with are the prime numbers (what about 1? I hear you cry; well, when you decompose 1 into factors you get the empty set of factors, all of whose elements are in fact prime).
And here are the actual values:
2 II JOINING | 501 DI ADHERING
5 V AGGRAVATE | 90 XC EXACERBATE
11 XI ANNEXING | 501 DI APPENDING
151 CLI WATCHLIST | 1 I REGISTER
7 VII VANQUISHING | 501 DI DEFEATING
59 LIX PLAINTEXT | 160 CLX CLEARTEXT
541 DXLI DESEXUALIZE | 1150 MCL EMASCULATE
1201 MCCI MOCCASIN | 51 LI SLIPPER
53 LIII LITIGATION | 51 LI LAWSUIT
1009 MIX MIX | 1 I STIR
401 CDI CODIFY | 50 L TABULATE
3001 MMMI METHAMPHETAMINE | 498 IID OPIOID
101 CI CONQUERING | 1 I SUBJUGATING
You may notice that the second-last one of those on the right
is nonstandard because you can't really put two smaller "decrementing" Is before a single D. DII is also composite (of course; it's even and doesn't equal 2) but if we just add up Roman numerals without regard to position then MIX no longer works because MXI = 1011 which, being a multiple of 3, is not prime.
But the questioner indicates in comments that
his intention was that anything not containing a valid Roman numeral should be a Not Decomposed Word, so OPIOID is fine for that reason.