Praise be to Allah, the Lord of the Worlds; and may His blessings and peace be upon our Prophet Muhammad and upon all his Family and Companions.
In principle it is permissible to eat the food of the People of the Book. Allah Says (interpretation of meaning): {….The food (slaughtered cattle, eatable animals, etc.) of the people of the Scripture (Jews and Christians) is lawful to you…..}[5:5].
But if they are known to leave out one of the conditions which allow us to eat their slaughtered meat, for instance if they devote it to other than Allah when slaughtering, or they do not cut from the animal what should be cut when slaughtering, or they kill it by stoning it to death, or drowning it etc., then it is not permissible for us to eat their meat, whether they write on it that it is slaughtered according to Islamic principles or do not write anything on it.
Allah knows best.