We know from Avengers
that Black Widow is a known element within Russian circles. Her "captor" makes reference to, paraphrase, "...that the Black Widow is nothing more than just a pretty face". Indicating she has some reputation but to what extent is unknown. I'd wager that reputations was earned while she worked for a Russian Black Ops outfit.
Natasha while "interviewing" Loki says, "I'm Russian, or I used to be regimes come and go..."
While talking to Hawkeye and he's talking about being "unmade" that he feels like his brain was scooped out and replaced and turns to Natasha and says, "You know what that feels like?" to which she responds, "You know I do."
Fairly certain she's referring to the Russian brainwashing/training.