Here's what David Crystal says about The gotten/got distinction in
The Cambridge Encyclopedia of the English Language (p.311):
"Gotten is probably the most distinctive of all the AmE/BrE grammatical
 differences, but British people who try to use it often get it wrong.
 It is not simply an alternative for have got.  Gotten is used in such
 contexts as 
    They've gotten a new boat.   (= obtain)
    They've gotten interested.   (= become)
    He's gotten off the chair.   (= moved)
 But it is not used in the sense of possession (= have).  AmE does not
   *I've gotten the answer.
or *I've gotten plenty.
 but uses I've got as in informal BrE.  The availability of gotten
 does however mean that AmE can make such distinctions as the following:
    They've got to leave  (they must leave) vs
    They've gotten to leave  (they've managed to leave)."
I'd add that Crystal's I've gotten the answer isn't starred if it means I have figured out the answer, rather than I have the answer.

The key is the overlap between the Possessive use of have and the Perfect use of have, plus the fact that one of the senses of get is come to have. If one has come to have a cold, for instance, then one has a cold, and the AmE usage of has got means that one is currently infested, due to the present relevance aspect of the Perfect. This is so common that kids regularly use got without have or even -'ve to mean have, and young kids even think it's the regular verb for possession, as witness such constructions as He gots new shoes.

Faced with the overwhelming interpretation of (ha)ve got as simply have, AmE has reinvigorated an old past participle gotten to be used whenever other, non-possessive forms of get are intended.

If one is simply speaking of the acquisition of something, for instance, rather than the current possession, one says I've gotten ..... in AmE since I've got implies that one still has it, and therefore focusses on the current Possession rather than the Perfective acquisition. And all of the idiomatic uses of get, like the get-Passive of get married, the Inchoative become/come to be inherent in get tired, the Concessive of get to go that Crystal mentions, etc. use gotten as their participle. Whereas any construction, even an idiomatic one like have to (= must) where one can use have equally well, use got as the participle.

Weird, but that's English for you.

followup --

>John doesn't disown the second example sentence, so I suppose he's
>one of the AmE speakers who can make that particular distinction.
>I certainly am NOT another; not only can I not imagine myself
>producing the second sentence (with any meaning), I had immense
>trouble understanding it (with any meaning) when I read it, and
>cannot imagine that I'd have understood it if anyone had spoken
>it to me.  (Do Crystal's example sentences come from a corpus,
>or does he make them up as he goes along?  If the latter,
>could he--a BrE speaker, I guess--just be flat out wrong in 
>this particular case?)  
He doesn't say, but I assume they're simply made as the need arises, like all sentences. It might not be the best of all possible example sentences -- this is just a sidebar in the Encyclopedia, after all -- but certainly one hears this sort of thing all the time, if one is paying attention. Consider:

I could also use got with the last one above (though not with the first two); however, it would have a slightly different sense, focussing on the resultant state, a paraphrase of

rather than the simple negation of the work with gotten.
  - John Lawler       Linguistics Department and Residential College     University of Michigan

    "Language is the most  massive  and  inclusive  art  we know,  a           - Edward Sapir
      mountainous and anonymous work of unconscious generations."       Language (1921)

More English Grammar   More About Language   The Eclectic Company   The Chomskybot