The schools agree on the principle; the split is over the slaughterer's tasmiyah

Surah al-Ma'idah states that the food of those who were given the Scripture is lawful for Muslims, and that Muslim food is lawful for them. Surah al-An'am prohibits eating that upon which Allah's name has not been mentioned. Both 5:3 and 2:173 forbid what has been dedicated to other than Allah, alongside carrion, blood and swine. The tafsir tradition reads "food" in 5:5 as their slaughtered animals, since other foods were already lawful by origin; Ibn Abbas is reported saying so.Quran.com+5

Aisha asked about meat brought by people recently out of disbelief, with no knowledge whether Allah's name had been mentioned, and the Prophet ﷺ told her to mention Allah's name over it herself and eat it. A Jewish woman brought him a poisoned roasted sheep and he ate from it. The Prophet ﷺ did not instruct asking the slaughterer whether he had named Allah.Sunnah.com+5

The instruction is also reported that whatever makes the blood flow, with Allah's name mentioned over it, may be eaten, except what is slaughtered with a tooth or a fingernail.Sunnah.com+1

The Hanafis hold their slaughter lawful in principle but make Allah's name a condition, so deliberate omission makes the meat haram whether the slaughterer is Muslim or kitabi; if he says "in the name of Allah" while intending the Messiah, the wording governs and it is eaten, but if one hears him mention the Messiah at slaughter, or say "the third of three", it is not eaten. The Malikis hold it lawful, though Malik strongly disliked what they slaughter for their churches and festivals; their preponderant position is karaha when a kitabi slaughters for a Muslim at his request, and what they slaughter as an offering to their gods is not lawful since it is not "their food".IslamQA.org+5

The Shafi'is hold it lawful, the slaughterer's qualification being that his marriage would be lawful; the tasmiyah is a sunnah, not a condition, so leaving it deliberately or forgetfully does not make the meat haram, but if he names other than Allah it is not lawful. The Hanbalis hold it lawful with no difference between a dhimmi and one at war; the tasmiyah is a condition when he remembers and is waived by forgetfulness, and deliberate omission or naming other than Allah makes it haram.Shamela+5

One line reads 6:121 as covering what was not consecrated to Allah at all, so the kitabi's tasmiyah remains a condition; the Permanent Committee cites 5:5 as the base and holds that if we know he named other than Allah it is not eaten, and if we do not know, the default is permissibility. Another reads 6:121 as aimed at carrion and what is dedicated to idols, not at a missing tasmiyah, as reported from Ibn Mas'ud and Ibn Abbas, with 5:5 making the exception for the People of the Book.IslamQA+3

A third distinguishes purpose: what is slaughtered for a church, a festival or a god is excluded from "their food," while what they slaughter to eat is included, provided Allah alone was named. A fourth leaves the unknown alone: since the Companions were told to say the name and eat rather than interrogate, most scholars do not require investigation of imported meat whose origin is a Christian or Jewish country.Shamela+5

Egypt's Dar al-Ifta holds the verse general, with nothing in it or in the hadith requiring inquiry into the method, and allows imported meat when the majority of the exporting country are People of the Book, unless it is established that the animal was not slaughtered, the slaughterer is not a kitabi, it is a forbidden animal, or it was killed by stunning or strangling. Ibn Baz allowed their slaughter today unless one knows it was by strangling, a blow, shooting or electrocution, and told a questioner who did not know whether Allah's name was invoked to invoke it himself.Dar al-Ifta (Egypt)+4

Ibn Uthaymeen held that their tasmiyah and a valid method are still required, but that meat from their countries is lawful without asking, and that as long as they claim affiliation with the two religions the rulings of the People of the Book apply to them. Al-Albani permitted it on the text of the Quran when the slaughter is proper and did not make the formula used a condition. AMJA answered that the distortion of their religions is not a modern development, since the Quran condemned it at the time of revelation and still permitted their food.Al-Fatawa+5

Stunning that kills before the throat is cut leaves carrion, which no school permits. Some Hanafi muftis today hold that most contemporary Christians and Jews no longer meet the description and that their meat should be avoided.IslamQA.org+4