It really depends on the background of the singer and how much exposure and training they have for different languages.
Can you imagine the number of countries and different people that cover American Classics? Songs like "The Greatest Love of All," "How Deep Is Your Love," and "My Girl," are always sung.
Take Korean pop for example. One singer by the name Bi/Rain sang "Just Once," in one of his concerts, and you could definitely hear his accent when it came to pronouncing the R.
Filipino artists such as Regine Velaquez can sing in English without any accent at all! Even though most Filipinos have accents when they speak English, the home accent is completely erased when they sing. It's the constant exposure to American culture.
One example that his close to home would be Celine Dion. She was taught to speak in English, and so her French accent can hardly be noticed by American listeners.
It's just exposure, really. The more exposed you are to the linguistic backgrounds of musical intonation, the more you try to accommodate it.