Islam has a law whereby a thief's hand is amputated, however Judaism does not proclaim such a punishment (Exodus 22).
What would happen to a Jewish thief under Shariah law? Would his hand be amputated? Or would he be punished pursuant to the teachings of his own religion?
The default rule is that Dhimmis are subject to all the laws of Islam, with only minor exceptions (personal religious matters such as the drinking of wine). Theft is not an exception but rather follows the default rule. If the Jew is a Dhimmi then his\her hand shall be cut.
Allah says in the Quran:
والسارق والسارقة فاقطعوا أيديهما جزاء بما كسبا نكالا من الله
[As for] the thief, the male and the female, amputate their hands in recompense for what they committed as a deterrent [punishment] from Allah.
This does not make an exception for non-Muslims rather its wording is general and inclusive of all thieves. And there are no exceptions for non-Muslims in ahadith either.
And the scholars have explicitly noted that the punishment of amputation for theft applies to Dhimmis just like it applies to Muslims:
الإسلام ليس بشرط فيقطع المسلم، والكافر لعموم آية السرقة
Islam is not a condition (for enforcing the punishment of theft), hence the hand of both a Muslim or a Kafir shall be cut because of the universality of the verse of theft
— Badaa'i' as-Sanaa'i' - (Hanafi)
ويقطع مسلم وذمي بمال مسلم وذمي
The hand of a Muslim or Dhimmi shall be cut for stealing the property of a Muslim or Dhimmi
— Minhaj al-Talibin - (Shafi'i)
وأما السارق الذي يجب عليه حد السرقة فإنهم اتفقوا على أن من شرطه أن يكون مكلفا، وسواء أكان حرا أو عبدا، ذكرا أو أنثى، مسلما أو ذميا
As for the conditions regarding the thief which are necessary for enforcing the Hadd of theft. They are in agreement that the thief be mukalaf regardless of whether he is free or a slave, male or female, Muslim or Dhimmi.
— Bidayat al-Mujtahid - (Maliki)
ويقطع المسلم بسرقة مال المسلم والذمي، ويقطع الذمي بسرقة مالهما. وبه قال الشافعي، وأصحاب الرأي، ولا نعلم فيه مخالفا
The hand of a Muslim shall be cut for stealing the property of either a Muslim or a Dhimmi, and the hand of a Dhimmi shall also be cut for stealing their property. And this is also the saying of Shafi'i and the Ashab al-Ra'y (Hanafis) and we do not know of any difference of opinion in this regard.
— Mughni Ibn Qudamah - (Hanbalii)
The Ottoman Empire operated a 'millet' system where people were judged under their confessional law. So there he would not be.
But even in Islamic countries today the vast majority of theft is punished by jail-time. So there it really wouldn't matter.
