The Bill of Rights, like most of the Constitution except clearly identified provisions, wasn't originally intended to apply to the states. It was only in the 20th century that the Supreme Court started the process of incorporating provisions of the Bill of Rights to the states via the equal protection clause of the 14th amendment (which was a reconstruction amendment explicitly imposing restrictions on the states).
So the clear text of the Constitution, as the drafters understood it, did not prohibit concurrent federal and state criminal trials for the same crime. Indeed, it imposed no limits on state criminal procedure at all. No due process, no nothing.