"These men do not eat meat."
Translation:Acești oameni nu mănâncă carne.
You could say, "Oamenii ăștia nu mănâncă carne," but I'm not sure if you could also say, "Ăștia oamenii nu mănâncă carne." Anyway, with ăștia the noun that follows it has to carry the definite article (the extra 'i').
Similarly to ăștia, if you were to use aceștia in its place, you would have: "Oamenii aceștia" or "Acești oameni"