Does double jeopardy prohibit prosecution, for the same event, in both federal and state court?
Key precedents on the question are US v. Lanza (1922) and Abbate v. US (1959). From Lanza:
The defendants thus committed two different offenses by the same act, and a conviction by a court of Washington of the offense against that state is not a conviction of the different offense against the United States, and so is not double jeopardy.
From Wikipedia's article on Abbate:
Abbate asked that the Court overrule its prior decision, which the Court declined to do. The Court reasoned that overruling Lanza would result in serious and undesirable consequences. Particularly, the state conviction here resulted in only three months' imprisonment, while the federal conviction made up to five years of imprisonment available. The Court deemed this potential disparity to be problematic. The only way to ensure that federal law enforcement interests would be vindicated under such a regime would be to displace state power to prosecute actions that also constitute federal crimes, which would be a massive shift in the balance of criminal power as between the states and the federal government.
In a comment, you raise a question about the last sentence:
"The only way to ensure that federal law enforcement interests would be vindicated under such a regime would be to displace state power to prosecute actions that also constitute federal crimes," seems to state that not displacing state power to prosecute, under such circumstances, would keep the law enforcement interests un-vindicated (which would mean unjustified). So the only justified thing to do would be to try under the Federal law and to not try under the state law.
"Vindicate" here does not mean justify but rather maintain or substantiate. The point is that if double jeopardy protected against prosecution by the federal government for a crime that has already been charged under state law, then states could effectively neuter federal crimes with which they disagreed by establishing a crime with identical elements and a token punishment. Another way of avoiding this outcome without permitting separate state and federal prosecutions for the same crime would be to prohibit state prosecutions altogether for acts that are also federal crimes. In other words, this says that the court found that the prohibition against double jeopardy allows these separate prosecutions because prohibiting them would require a "massive shift" in the responsibility for enforcing criminal law away from the states and toward the federal government.
(Of course, this works both ways; if double jeopardy applied across sovereigns then the federal government could also neuter state laws. More generally, the current approach to double jeopardy may be seen as maintaining the balance of power between the states and the federal government, more than as protecting the interests of one side over those of the other.)