Not neccessarily. It really depends on the translation. The Douay Rheims represents, to a large extent, a translation from the Latin Vulgate. The beauty of the Peshitta and the Vulgate is they represent early translations from the Greek, and other sources in the case of the OT; they provide a frame of reference.
That said, one should not accept the Aramaic Primacy hypothesis; the Greek NT texts are original. Ehat is more, the Peshitta is certainly not, as Lamsa argued, the original text of the NT. The Aramaic Source Hypothesis argued by
@SteveCaruso is very helpful however.
Note that I am Syriac Orthodox; my church actually uses the Peahitta as its main liturgical text. The Western Peshitto as it is known features all of the books of the Athanasian canon, and there is a rather nice 19th translation of it by Murdoch.