You never notice when someone speaks another language with an accent, amirite?

My favorite is French spoken with a Texan accent.

I can notice the accent of a french or a chinese when they talk in English. And my native language is Spanish.

My ex spoke kosraen, an island language and he and his brothers favorite accent to try to do was like Georgia country accents for some reason ( we worked at a bar w a bunch of redneck regulars) and hearing them attempt to sound country and fail miserably was the best

I can't understand their mouth sounds.

Depends on the other language and on the accent.
 
I certainly can  tell if someone is speaking Spanish or German with an American accent.
I don't think I could tell with, say, Farsi, however.