It's not very clear actually because there were no rules regarding that. (AFAIK) We only have records for official Kadis(Regional Judges) and Divan-i Humayun (higher court). There are cases seen at kadi or divan-i humayun where one side is Muslim and the other side non-Muslim.
They are both institutions of the state and since the Ottoman Empire was an Islam state, they were obviously Islamic. But, Divan-i Humayun's decisions were "Sultan's Word" so they had to be fair. The usual/accepted route is to go Kadi first and then you can appeal to Divan-i Humayun. But it's seen im records that sometimes non-Muslims directly went to Divan-i Humayun because they believed that they wouldn't be treated fair at Kadi.
The judicial system of this empire was quite accommodative for an Islamic system. It had Islamic courts which were the primary courts and they formed the cornerstone of administration in the empire. But there were also other courts that were not based on Sharia law and were meant for the interpretation of the law as understood by non-Muslims who were part of the empire. They included Jews and Christians who had been welcomed to the empire and lived in groupings called millets. But the Jews and Christians had the freedom to take their cases to the Islamic courts if they wanted to and many did as a way of giving judicial credence to their cases, given the immense unofficial respect that was given to the Islamic courts. It should be noted that this recognition was informal and any decision from the other courts were equally respected in the empire.
The third type of courts was the trade/commercial/industrial or business courts. In these courts, disputes emanating from business within the empire were solved. The parties involved determined the inclination and sometimes, both Islamic law experts and Christian and Jewish judicial officers would sit side by side and decide a case. They were more concerned with securing justice and fairness and religion was never a key consideration. New laws into the three types of courts came from the top leadership as well as the judicial experts of the three courts.
Do you know how this worked with disputes between people of different religions?