Gemara

24b (Mishnah): If an animal gored, pushed, bit, crouched or kicked in the victim's premises, R. Tarfon obligates full damage;

Chachamim say, it pays half-damage.

R. Tarfon: In a Reshus ha'Rabim, the Torah is lenient to exempt Shen and Regel, but Keren pays half-damage. In the victim's premises, where Shen and Regel pay full damage, all the more so Keren pays full damage!

Chachamim: Dayo (we cannot learn from a Kal va'Chomer more than the source). It pays only half-damage, like in a Reshus ha'Rabim.

47a (Mishnah): If Shimon brought his pots or fruit into Reuven's yard without permission, and Reuven's animals broke or ate them, he is exempt;

If Reuven's animals were damaged by the pots or fruit, Shimon is liable. If Shimon had permission to bring them in, Reuven is liable.

If Shimon brought his ox into Reuven's yard without permission, and Reuven's animals damaged it, he is exempt. If the ox damaged Reuven's animals, or fell into a pit and dirtied the water, Shimon is liable. If Shimon had permission to bring his ox in, Reuven is liable.

Rebbi says, in all cases one is liable only if he accepted responsibility for the other's property.

Inference: If Shimon had permission, he would be exempt when Reuven's animals were damaged. We do not say that Shimon accepted to guard Reuven's animals. This is like Rebbi, who says that Stam, one does not accept responsibility for another's animals.

Question: The middle clause obligates Reuven when Shimon had permission to bring his pots in. This is like Chachamim, who say that Stam one accepts responsibility to guard himself from damaging another's property. The Mishnah concludes with Rebbi's opinion. Can the Reisha and Seifa be like Rebbi, and the middle is like Chachamim?!

Answer #1 (R. Zeira): Different Tana'im taught the Mishnah. (The Tana of the middle clause holds that Chachamim argue with Rebbi. The Tana of the Reisha and Seifa holds that Chachamim agree with Rebbi.)

Answer #2 (Rava): The entire Mishnah (until it mentions Rebbi) is like Chachamim. When Reuven gives permission, he accepts to guard Shimon's property even from falling in the wind!

48b (Rava or Rav Papa): If two people were both acting in a permitted way or both in a forbidden way and Heziku each other, they are liable. If Huzku (they got hurt) through each other, they are exempt.

(Inference): This is because they were both the same. If only one had permission, he is exempt and the other is liable.

Rishonim

The Rif (11a) brings the beginning of our Mishnah and (21a) the teaching of Rava or Rav Papa.

Rambam (Hilchos Nizkei Mamon 1:7): A Mu'ad pays full damage and a Tam pays half in the victim's Reshus. If the victim entered the damager's Reshus and the Ba'al ha'Bayis' animal damaged him, he is totally exempt, for he can say 'had you not entered, you would not have been damaged.' It says "And he will send his animal and it will consume in another's field."

Rambam (7:4): If David brought his ox into Levi's Chatzer without permission, and Levi ox gored it or Levi's dog bit it, he is exempt. If David's ox gored Levi's ox, it pays half-damage, just like for goring in Reshus ha'Rabim.

Rambam (7:6): If David brought his ox into Levi's Chatzer without permission, and it damaged the Levi or Levi was damaged by it, or it dug pits in the Chatzer, David pays for damage to the Chatzer. Levi pays for damage of the pit, for he should seal it.

Rambam (7): If Levi unknowingly damaged the ox he is exempt, for he says 'why did you enter without my permission? I did not know until I stumbled upon you.' If he knowingly damaged the ox he pay full damage, for he may expel it from the Reshus, but he may not damage it.

Rambam (Hilchos Chovel u'Mazik 6:3): (If one damaged another's money, he pays full damage.) This refers to in the victim's Reshus. In the damager's Reshus, he is liable only if he damaged with intent. If he was Shogeg or Mezid, he is exempt. If both had permission or neither had permission and one damaged the other without intent, he is exempt.

Rebuttal (Ra'avad): This is not clear. Even if both had permission or neither had permission, if one Hezik, he is liable. If Huzku through each other, they are exempt.

Magid Mishneh: The Rambam exempts one who damages in his own Reshus, even if the victim had permission to enter. He holds that Heziku is with intent, and Huzku is without intent. The Ra'avad holds like Rashi.

Beis Yosef (DH u'Mah she'Chosav Hayu): Rashi explains that a case when both had permission is when if the Ba'al ha'Bayis allowed Ploni to enter his Reshus. They are liable only for overt damage. The Rambam obligates even for Shogeg or Ones in the victim's Reshus.

Poskim

Shulchan Aruch (CM 378:4): (If one damaged, he pays full damage, whether he was Shogeg or Anus.) This refers to in the victim's Reshus. In the damager's Reshus, he is liable only if he damaged with intent, like we will explain.

Shulchan Aruch (6): Anyone who damages must pay, whether in Reshus ha'Rabim or the victim's Reshus. Even in the damager's Reshus, if David entered Levi's Reshus without permission and Levi damaged David's body or property, he is liable.

Shulchan Aruch (7): If both of them had permission or neither had permission and they were damaged through each other's body or property, if they did not know that the other is there, they are exempt. If they saw each other, even though they did not intend, they are liable.

Gra (19): The Shulchan Aruch mentions body and property to oppose the Rambam. One can explain that regarding damage to people, the Rambam obligates for overt damage even without intent, but for damage to property he obligates only with intent. The Magid Mishneh and Lechem Mishneh say that the Rambam equates damage to property and to people.

Shulchan Aruch (398:1): If David brought his ox into Levi's Chatzer without permission, and Levi's ox gored it or Levi's damaged it he is exempt, for he says 'since you entered without my permission, I did not know until I stumbled upon it.'

Shulchan Aruch (2): If David's ox gored Levi or Levi's ox, a Tam pays half damage and a Mu'ad pays full damage, just like for goring in Reshus ha'Rabim.