When people imagine Natasha's voice in the comic books, do you hear a Russian accent when she speaks English? In most animated adaptations I've seen, she has an accent. However, Scarlett's interpretation in the movies does not.
Initially, I preferred the Russian accent, but after thinking about it, I think it makes more sense if she didn't have one. As a spy, I assume she would have been taught to speak every language in her repertoire with its respective native accent, since a Russian accent would be too conspicuous.