As long as -- seemingly -- the adverb "quite" in AmEng idiomatically carries an emphatic sense to it -- pretty much similar to saying "completely" or "absolutely" as in, "That girl looks quite pretty!" -- what adverb (or phrase, or grammatical construction) would Americans typically use -- in speech and writing -- that would coincide most with the chiefly BrEng meaning of "quite" [=to a noticeable or a partial degree]?
http://dictionary.reference.com/browse/Quite?s=t
http://dictionary.reference.com/browse/Pretty?s=t
http://dictionary.reference.com/browse/Rather?s=t
http://dictionary.reference.com/browse/Fairly?s=t
http://dictionary.reference.com/browse/Somewhat?s=t
Would they more likely say something like:
That girl looks somewhat pretty.
That girl looks rather pretty.
That girl looks fairly pretty.
That girl looks not too [...].
That girl looks pretty to some degree (or extent, or point).
Etc.
Or, would they rather use a "-ly" ending adverb and say:
That girl looks moderately pretty.
That girl looks passably pretty.
That girl looks partially pretty.
Etc.
In informal contexts, would they more likely say something like:
That girl looks sort of/kind of (sorta/kinda) pretty.
That girl looks some pretty.
That girl doesn't look too [...].
Etc.
Consider the phrase, "That girl looks...pretty" merely as an example. The adverb (or phrase, or construction" should also work with other examples of BrEng "quite", e.g. "I'm quite well".