"These men do not eat meat."

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"

Here are the correct translations in Romanian:
Acești oameni nu mănâncă carne. (the original translation)
Oamenii aceștia nu mănâncă carne. (the second version)
Oamenii ăștia nu mănâncă carne. (ăștia is just an informal shortening of aceștia)