Bava Kamma, Chapter 4, Mishnah 3. This mishnah returns to and expands upon two points that were already taught at the beginning of the tractate, in Chapter 1, Mishnah 2, where it was established that the laws of damages apply exclusively to two types of property:
"Nechasim she'ein bahem me'ilah" - Property that is not subject to the laws of me'ilah, meaning property that does not belong to the Beis Hamikdash.
"Nichsei bnei bris" - Property belonging to a fellow Jew.
Our mishnah comes to examine and expand upon these two laws.
Damage between a Jew and Hekdesh:
We learned in the mishnah: "Shor shel yisrael shenagach shor shel hekdesh, veshel hekdesh shenagach leshor shel yisrael - patur" - An ox of a Jew that gored an ox of Hekdesh, or one of Hekdesh that gored an ox of a Jew, he is exempt. Whether the animal of a Jew gored an animal belonging to Hekdesh, or the animal of Hekdesh gored the animal of a Jew - there is no payment in either direction.
The source of this law is in the verse (Exodus 21:35): "If a man's ox gores the ox of his fellow." From the word "his fellow" it is derived that the laws of damages apply specifically to the animal of a fellow Jew, and not to an animal belonging to the Beis Hamikdash, which is not included in "his fellow".
Although the verse deals with a tam that caused damage with its horn, the law applies equally to all types of damage - mamon, as well as pit, fire, and the like. There are no damage payments paid to Hekdesh or out of property belonging to the Beis Hamikdash.
Damage between a Jew and an idolater:
The mishnah continues: "Shor shel yisrael shenagach leshor shel oved kochavim - patur" - An ox of a Jew that gored an ox of an idolater, he is exempt. The Jew pays nothing, whether the ox is a tam or whether it caused damage classified as mamon, and the like.
According to the Bartenura, the source of this law is a drashah from a verse in Habakkuk, meaning that this is a rabbinic law: "He stood and measured the earth; He looked and released the nations." On a simple level, Hashem stands and measures the earth, and the nations of the world tremble before Him; but by way of drush, "vayater" (released) is related to the word "hatarah" (permitting), meaning that He permitted their property and revoked their claim to payment.
In the context of the laws of damages, a Jew does not pay an idolater for damage caused to his property. The basis for this is a sort of fine for their not fulfilling the seven Noahide laws as they are obligated to, and therefore they are fined by Israel in that they do not collect damage payments when they suffer damages. The Meiri explicitly states that this stems from the non-fulfillment of the seven Noahide laws, but a nation that fulfills the seven laws - which arguably includes Muslims, Christians, and the like - is entitled to its payments as usual.
From the other side, we learned: "Veshel oved kochavim shenagach leshor shel yisrael, bein tam bein mu'ad - meshalem nezek shalem" - And one of an idolater that gored an ox of a Jew, whether it is a tam or a mu'ad, he pays full damages. There is no practical difference whether the animal is a tam or a mu'ad, and in any case the gentile pays full compensation to the Jew, and not the usual payment of half damages for a tam. The reason is that same foundation of "He permitted their property to Israel", and as the Rambam explains, this strict obligation comes as a fine, so that they should be more careful and guard their animals from causing damage.
Therefore, according to the Meiri, if they had established courts and a judicial and enforcement system, as required by one of the Seven Noahide Laws to ensure law and order - they would not be liable for more than half damages for a tam ox.
In summary: This mishnah expanded upon two of the rules taught in Chapter One - that the laws of damages only apply to property not subject to the laws of me'ilah, and only to the property of members of the covenant. Regarding damages involving consecrated property (hekdesh), there is an exemption in both directions, derived from the exposition of "the ox of his fellow" - excluding an ox belonging to hekdesh, and this applies to all types of damage. Regarding damages involving a non-Jew, derived from the exposition of "He looked and startled the nations," a Jew who caused damage is exempt, whereas a non-Jew who caused damage pays full damages whether the animal was a tam or a mu'ad - a penalty for their failure to fulfill the Seven Noahide Laws, and according to the Meiri, this penalty does not apply to a nation that fulfills them.