Apparently many, but not all, sources hold that it is a mitzvah to relieve animals of their suffering, whether or not the animal belongs to you. See this passage from a book by R’ Dovid Sears:
The Chasam Sofer advances the opinion that, although causing pain to an animal is scripturally forbidden, the obligation to rescue an animal in distress applies only to one’s own animals; see Teshuvos Chasam Sofer, Yoreh De’ah, nos. 314, 318, and Choshen Mishpat, no. 185. This appears to be consistent with the view of Maharam Schick on the Taryag Mitzvos, Mitzvah 80. Authorities who do not make such a distinction include: Noda B’Yehudah, Mahadurah Kama, Yoreh De’ah, nos. 81-83; Shulchan Aruch HaRav, Vol. 6, Tza’ar Baalei Chaim, 4; Kitzur Shulchan Aruch, 191:1; Orach Maysharim, 15:1; et al. The Netziv in Ha’amek Davar, Deuteronomy 22, maintains that one is not obligated by Torah law but is required to intervene by rabbinic decree. In Eishel Avraham: Tinyana, Yoreh De’ah 305:20, R. Avraham of Butchatch argues that the relief of tza’ar baalei chaim directly or indirectly caused by a human being is incumbent upon any Jew capable of intervening by scriptural law. For further discussion see R. Yitzchak Nachman Eshkoli, Tza’ar Baalei Chaim (2002), chap. 11.