The decision will be made by whichever country arrests him first (although a minority of countries allow for the trial of people who break their criminal laws in absentia). Needless to say, if nobody manages to arrest him ever, he will not face any criminal consequences except the issuance of an arrest warrant possibly accompanied by a pre-existing conviction in absentia if arrested in countries that allow for such a proceeding.
Normally, in these circumstances, either country would have jurisdiction under its own laws to prosecute and punish the criminal, and many extradition treaties would not require the extradition of someone who committed a crime punished domestically in the state in whose custody the criminal is as part of the same course of events.
Many countries will not extradite someone if they could face the death penalty in the receiving country. But, sometimes law enforcement in a country with a less serious penalty will intentionally defer to law enforcement in a country with a more serious penalty that is simultaneously trying to arrest him.
Ordinarily, law enforcement is not authorized to use deadly force to arrest someone who is simultaneously being arrested by law enforcement from another country against either the arrestee or the law enforcement from the other country. Indeed, using deadly force against another country's law enforcement officers who are carrying out a lawful arrest in their own country would ordinarily be considered an act of war.
U.S. double jeopardy provisions of the constitution do not prohibit a second prosecution of an offender in these circumstances because of a first prosecution by another sovereign, but many prosecutors in many countries would decline to prosecute someone a second time for the offense that they have already been convicted of in exercise of their discretion, and many judges would consider time served in another country for the same offense as a factor in setting their own sentence.