I believe it doesn't work like that for a language that is not your first. I can't hear my parisian accent when I speak in french, but I can definitely hear my french accent when I speak in english, though I've been told my parisian accent is worse.I would imagine they don't notice it at all. I was told once by my friends in Chicago that I have an "Indiana accent" to which I said "I don't have an accent. Y'all got the accent"
This sequence from president Chirac visiting Jerusalem in 1996 has become totally cult in France because of his accent :How do French people perceive their accent when speaking English, and how do they view the French language's romanticism in comparison to other languages?
I wouldn't know about language vs romanticism, some very harsh sounding languages such as german produced great romantics...
We have had some ok bands singing only in english, like Yodelice, who became known with this song.