Can the Islamic State punish a non-Muslim citizen who commits offense against a Muslim? For example if a non-Muslim citizen steals a Muslim's property, will the culprit's hand be cut? Or if a non-Muslim citizen commits adultery with a Muslim citizen, will the non-Muslim culprit be sentenced to death (rajam) along with the Muslim culprit?
The shari'ah is silent on this issue.
In this regard, both ways can be adopted depending upon the terms of citizenship concluded with the non-Muslim minority: A Muslim state can enforce Islamic punishments on the non-Muslims citizens, and it can give them the leverage to follow their own code in a particular matter.
The bottom line is that non-Muslims citizens are not necessarily bound to face the punishments prescribed by the Islamic shari'ah.