Answer
Wa `alaykum as-salamu wa rahmatullahi wa barakatuh.
In the Name of Allah, Most Gracious, Most Merciful.
All praise and thanks are due to Allah, and peace and blessings be upon His Messenger.
Dear sister, we really appreciate your question, as each Muslim has to fully observe his/her foods, especially in Muslim-minority countries, making sure that it is lawful (halal).
As for your question, it is a controversial issue. Some juristic schools see that eating the meat of People of the Book is permissible if they slaughter in the same way as ours and mention the name of Allah at the time of slaughtering. Other scholars permit it without any condition. So, we should not accuse each other and let each one act upon the view which he/she feels more comfortable with.
In his response to your question, Prof. Dr. Monzer Kahf, a prominent economist and counselor, states:
Explaining the whole issue requires several lectures but I will try to give a very brief summary:
The Qur’an and the Hadiths came in Arabic language; these are the sources of our Shari`ah, no doubt about it. These texts very often have several meanings in many miner issues. This is an expression of the mercy of Allah that He left many details to our understanding.
The issue of meat is one of them; the Qur’an permits eating the meat of ahlul-lkitab (the people of the Book, i.e., the Jews and Christians). Some scholars took it with the condition that they slaughter in the same as our way and mention the name of Allah at slaughtering while some others took it without any conditions. Let us trust each other and let anyone select the view he/she feels more comfortable with, as long this view is authentic.
In fact, this view that she expressed is not the view of Ibn `Uthaymeen himself. It is the known view of the schools of Shafi`i and Hanbali Schools. What is much more important though is not to allow such differences negatively affect our sisterhood/brotherhood. So that we continue love each other, respect each other and live together.
If you eat halal according to the Hanafi view, she eats halal too according to the Shafi`i view and if she invites you and offer food in her way, you should accept it and vice versa.
We must remove such frictions from our relations and realize brotherhood and sisterhood through diversity rather than conformity. Following any of the respected views in fiqh is not better or worse than following another respected view in fiqh.
Allah Almighty knows best.