As regards Pennsylvania Dutch German and German German, my German German acquaintance overheard Pennsylvania Dutch German speakers and claimed they were incomprehensible to her, I’m not sure how hard she tried.
Right there is why children are so much better at learning languages than adults. Children think it’s fun to mimic, and they have a natural talent at it. But when they mimic someone who talks funny, their parents and teachers are horrified, mortified, and scold them not to do that. So we all grow up with heavy inhibitions about mimicking.
But that very activity of young children is precisely what it takes to learn to speak a language accurately. As an adult, I’ve had to consciously decouple my shame and inhibition about mocking others and approach it with the innocence of a young child. People ask me how did I learn so many languages so easily. Well, this had a lot to do with it. We adults still have that ability, if only we don’t fight it and suppress it. We just have to distinguish between legitimate (language learning) and non-legitimate (mockery) ways of applying it.
Just make sure you mimic real native speakers of the language and avoid the stereotyped accents used by comedians. I.e. can the “stage Irish” and the spicy meatballs.
Kissingers’s brother has a great quote attributed to him, possibly apocryphal. Supposedly, when asked why Henry has such a strong accent and he had no noticeable accent, he said, “I listen to people and Henry doesn’t.”
“Your mother likes to ride my airplane?” 
Like I said it’s pretty close to meaningless in English too. Basically a silly “Your Mom” joke.
I once tried to tell someone in Chinese, “your mom’s like a bus, cause guys are always getting on and off her all day” – but I only left them puzzled.
i first learned Spanish in the 1960s, so I guess that explains it. The Spanish lady who taught me took huge trouble to give me that soft “l” overtone in “ll”. She said only americanos pronounced it like “y” - and she meant that to be an insult. So I was surprised to be attacked for it here.
English speech is increasingly Americanised because of the impact of US media, so it’s not surprising if the same is happening to Spanish.
Quebec French also uses different profanity than Metropolitian French. And Quebec French also uses more feminine endings for professions rather than refering to women with masculine nouns (ie l’avocate, la docteure, la travailleuse, l’ambassadrice, etc).
I don’t recall any attacks, but even if you’d learned Spanish later, often language teachers are more picky than regular speakers - or even, in the case of languages with Academies, than those venerable institutions. They often try to teach some sort of “Standard [insert language here]” which will rarely if ever be found on the street. To wit:
- teachers of Spanish who teach “Valladolid Spanish minus its grammatical defects” (what they call Castillian),
- my cousin’s teachers of Catalan, who weren’t Catalan themselves and who could rant for hours on the difference between two kinds of S - a difference they were completely unable to pronounce; one of them was the kind of Andalusian who turns 90% of S into Z (in either Spanish or Catalan),
- teachers of Basque who teach and accept only Batua, an artificial dialect invented for the purpose of being the standard,
- or all those teachers of English who try to teach Her Majesty’s English to people who are a lot more likely to lay hands on US-accented movies and songs than on BBC series (plus, BBC series and British movies have people with all kinds of British accents, the people in Eastenders or Dr Who don’t all use Received Pronunciation).
OP here; sorry to have neglected my own thread(!). Thanks for the many interesting responses. Most of you seem to have understood what I was asking, but for those for whom there is lingering confusion:
I am not asking if there is an accent when an American whose first language is English learns a foreign tongue - clearly there almost always is. What I wanted to know is how/if non-English language groups in the US have formed their own, distinct accents that are audibly different from the language as spoken in any home countries. So, for example, has the Mandarin spoken in long-established American Chinatowns branched off to the point where a person born in one and raised speaking Chinese is recognizably speaking an accent of the language that tags him as American Chinese, as opposed to someone from Beijing.
Sounds like that is the case for a lot of language…kind of cool!
Historically, the Chinese communities in the United States spoke Cantonese, so you’d want to know if their accent was different from someone from Kwangtung, rather than someone from Peking.
I remember reading that many of the Italian Americans stationed in Italy during WWII were fluent in Italian, having learned from their families… but could not speak Italian with the locals due to dialect differences. Evidently it had something to do with linguistic changes in Italy and the US over a few generations.
Might also be that Italy has a lot of regional languages (although today most Italians know standard Italian), so perhaps those Italian-American soldiers just weren’t stationed in the right region.
Piedmontese is no more intelligible to Sicilians than French is. And vice versa.
Uh, yeah, quite a bit. Quebecois is usually fairly intelligible to a native Frenchman, but Cajun French may not be. I have a good friend from Louisiana who has spent about a year studying in France; she claims that backwoods Cajun French resembles French words constructed on an English syntax. Kind of.
So Cajun French is probably a good example of what the OP’s looking for-- Cajun French represents a drastic regional shift in a non-English language in America.