Only when the state statute coincidentally matches the federal statute, for crimes like bank robbery that are defined at the state and federal level.
If the state doesn't have a law against the crime (i.e. South Carolina has no laws regarding hate crimes [1], but the federal government does [2]), then federal prosecutors are responsible for prosecuting those. One can't escape prosecution for a federal crime because there is no state law against it. Prosecuting federal crimes at a local level is exactly why the regional US attorneys offices exist.
In the case in from the article, the crimes committed by the state Senator (wire fraud, money laundering, etc) are federal by definition, so a local DA isn't going to prosecute them anyways.
1. https://apnews.com/article/ahmaud-arbery-columbia-south-caro...