According to the piskei halacha, the butei dinim retain jurisdiction over klal yisroel even if chutz l’aaretz, in non-Jewish lands. Before the enlightenment this was recognized by most European governments which authorized the kehila to enforce crime and punishment in the Jewish community. This halacha didn’t change with the enlightenment and thus we aren’t permitted to turn our own over to the secular government.
Rav Moshe in Orach Chaim 5-9,11 and Choshen Mishpat 1-8 says you can’t support or have a Jew punished by the secular authorities because they administer a punishment in excess of halacha. So to answer your question, if a Jew steals his punishment should be as specified in halacha.
The sheva mitzvos mandating the goyim establish a court system applies to them to establish it to administer justice among themselves. They do not, in the eyes of halacha, obtain jurisdiction of the Jews in their lands.
(Any response to this comment should include addressing the psak halacha from Rav Moshe.)