What the Contemporary Scholars say...

Shaykh Abdul Aziz aal ash-Shaykh (Current Grand Mufti of Saudi Arabia)
 As the Grand Mufti of Saudi Arabia, he also serves as the Head of the Permanent Committee for Islamic Research & Fataawa (اللجنة الدائمة للبحوث العلمية والإفتاء)
When asked about the ruling on meat slaughtered in a country mixed with "Christians, idol-worshippers, and ignorant Muslims" and the uncertainty if the name was Allah was pronounced, he replied,

Praise be to Allaah.
If the matter is as described, with the meat being slaughtered by a mixture of people that includes people of the Book, idol-worshippers and ignorant Muslims, and you cannot tell who has slaughtered what and it is not known whether the name of Allaah was mentioned at the time of slaughter or not, then it is haraam to eat the meat when you are confused about who did the slaughtering, because the basic principle is that animals of the an'aam class (i.e., camels, cattle, sheep and goats), and animals that come under the same rulings, are haraam unless they are slaughtered in the proper manner as prescribed in sharee'ah. In this case there is doubt as to whether the slaughtering is done in the proper shar'i manner or not, because of the mixture of slaughterers, some of whose meat is permissible and some is not permissible – such as the idol-worshippers and innovators among the ignorant Muslims whose bid'ah involves shirk.
Al-Lajnah al-Daa'imah, 22/450-451.1

Shaykh Muhammad Nasiruddin al-Albaani
When asked about the permissibility of asking the host about their food and drink, he replied,

Praise be to Allaah.
It was narrated from Abu Hurayrah that the Messenger of Allaah (peace and blessings of Allaah be upon him) said: "When any one of you enters upon his Muslim brother and he offers him food, let him eat from his food and not ask about it. And if he offers him something to drink, let him drink what he offers and not ask about it."
(Narrated by Ahmad, 8933).

The Shaykh commented:
The apparent meaning is that this hadeeth is to be interpreted as referring to a case where a person thinks it most likely that his Muslim brother's wealth is halaal and that he avoids forbidden things. Otherwise it is permissible, indeed obligatory, to ask, as in the case of some Muslims who have settled in kaafir countries. In their case and in similar cases, we have to ask them about their meat, for example whether it is killed (by non-halaal methods) or slaughtered (according to sharee'ah).
Al-Silsilah al-Saheehah, Hadeeth 627.2

Shaykh Muhammed Salih Al-Munajjid
When asked about the meat of "People of the Book," he replied,

The scholars agree unanimously that it is permissible to eat the meat of the People of the Book, Jews and Christians, if the name of Allaah is mentioned at the time of slaughter, as Allaah says (interpretation of the meaning): "Eat not of that (meat) on which Allaah's Name has not been pronounced (at the time of slaughtering of the animal)…" [al-An'aam 6:121].
It is also a condition that the meat should be slaughtered in the manner prescribed by sharee'ah. If it is known that the slaughter was not done in the proper Islamic manner, e.g., by strangulation or electric shock and so on, then it is haraam.3

Shaykh Saalih Al-Fawzaan
Member of Permanent Committee for Islamic Research & Fatawaa
In providing the evidence for the requirement of tasmiyah (reciting the name of Allah) both for "People of the Book" (Christians & Jews) as well as Muslims, he quotes Imam Ibnul-Qayyim,

"Moreover the aforementioned noble verse indicates that the slaughtered animal is not lawful to be eaten if the Name of Allah is not mentioned when slaughtering it, even if the slaughter is a Muslim." 4

Shaykh Muhammad ibn Adam
Author, Director of the Institute of Islamic Jurisprudence (Darul Iftaa), England.
When asked about the meat commonly sold in Western stores, he replied,

Then there is another aspect to this issue, which is that, many people in the West claim to be Christians. This claim is not true insofar as they do not firmly believe in God, a revealed book, and a prophet, regardless of the nature of their beliefs in these matters. Rather, they are often Christians only by name and do not actually believe in any religion. Thus, they cannot be termed People of the Book. They are atheists and it is not permissible to consume the animal slaughtered by them, by scholarly consensus.

In conclusion, the meat sold in western supermarkets or served by Christians will be unlawful as they generally do not fulfill the conditions of a valid slaughtering and that they are usually not true Christians.5

Shaykh Abdul Aziz Bin Baz
Former Grand Mufti of Saudi Arabia, Former President of the Permanent Committees of Religious Research and Ifta.
When asked specifically about the meat in America, he replied,

If the region where the meat mentioned is found has only People of the Book, namely Jews and Christians, then their meat is halaal, even if it is not known how they slaughtered it, because the basic principle is that meat slaughtered by them is halaal, because Allaah says (interpretation of the meaning):

“Made lawful to you this day are At‑Tayyibaat [all kinds of Halaal (lawful) foods, which Allaah has made lawful (meat of slaughtered eatable animals, milk products, fats, vegetables and fruits)]. The food (slaughtered cattle, eatable animals) of the people of the Scripture (Jews and Christians) is lawful to you and yours is lawful to them” [al-Maa’idah 5:5].

If there are other kaafirs in the region, then do not eat it, because that means there is doubt as to whether it is halaal or haraam. Similarly if you know that those who sell these meats slaughter the animals in a way that is different from the shar’i method, such as strangling or electric shock, then do not eat it, whether the one who slaughtered it is a Muslim or a kaafir, because Allaah says:

“Forbidden to you (for food) are: Al‑Maitah (the dead animals — cattle — beast not slaughtered), blood, the flesh of swine, and that on which Allaah’s Name has not been mentioned while slaughtering (that which has been slaughtered as a sacrifice for others than Allaah, or has been slaughtered for idols) and that which has been killed by strangling, or by a violent blow, or by a headlong fall, or by the goring of horns — and that which has been (partly) eaten by a wild animal — unless you are able to slaughter it (before its death)” [al-Maa’idah 5:3].
Majmoo’ Fataawa Ibn Baaz, 23/20.6

Sources:
1 http://islamqa.com/en/ref/12569/slaughter.
2As quoted by Sheikh Muhammed Salih Al-Munajjid, http://www.islam-qa.com/en/ref/21661/meat.
3http://islamqa.com/en/ref/3261/people%20of%20the%20book.
4Fawzan, Salih, Dr.  “A Summary of Islamic Jurisprudence Volume II” pg. 664.
5http://www.daruliftaa.com/question.asp?txt_QuestionID=q-14415786.
6http://www.islam-qa.com/en/ref/127959/slaughter.