Narrated `Aisha: A group of people said to the Prophet, "Some people bring us meat and we do not know whether they have mentioned Allah's Name or not on slaughtering the animal." He said, "Mention Allah's Name on it and eat." Those people had embraced Islam recently.
Hadith Referenceصحيح البخاري / كتاب الذبائح والصيد / 5507
Hadith Gradingمحدثین:أحاديث صحيح البخاريّ كلّها صحيحة
Hadith Commentary: Some individuals have deduced from this hadith that reciting "Bismillah" over the slaughtered animal is not obligatory, because if it were obligatory, the Messenger of Allah (sallallahu alayhi wa sallam) would not have permitted the Companions (radi Allahu anhum) to eat the meat. However, this deduction is not correct, because although the Bedouins had only recently entered Islam, they were not ignorant of reciting "Bismillah." It is quite possible that they used to slaughter while reciting "Bismillah," but the Companions (radi Allahu anhum) expressed doubt about them, thinking perhaps they did not recite "Bismillah." So the Messenger of Allah (sallallahu alayhi wa sallam) removed their doubt, stating that one should have a good opinion regarding a Muslim. When they are Muslims, then certainly they would recite "Bismillah." However, to remove your own doubt, you should recite "Bismillah" yourselves. In this regard, a hadith is also presented: "The slaughter of a Muslim is lawful, whether he mentions the name of Allah or not." But this narration is weak in terms of its chain of transmission. (Irwa’ al-Ghalil: 8/169, no. 2537, and Fath al-Bari: 9/787)
Source: Hidayat al-Qari: Commentary on Sahih Bukhari, Urdu, Page: 5507
'Aishah (RAA) narrated, 'Some people said to Allah's Messenger (ﷺ), There are people who bring us meat and we do not know whether or not they have mentioned Allah's name over it.' He replied, "Mention Allah's name yourselves and eat it." Related by Al-Bukhari.
➋ Regarding the People of the Book (Ahl al-Kitab), if it is certain that he mentioned the name of Allah while slaughtering—for example, one witnessed the slaughter himself or a trustworthy Muslim witnessed it—then the animal slaughtered by such a person from the People of the Book is also permissible.
The slaughtered animal of other non-Muslims is not permissible.
And Allah knows best.
Source: Bulugh al-Maram: Commentary by Safiur Rahman Mubarakpuri, Page: 1151