Sodium ion Na+ should not be confused with metallic sodium Na. They are actually two different chemical species (atoms vs ions), so it is expected that they chemically behave in a different way. The same applies to chlorine Cl and chloride ion Cl-.
Na can "switch" to Na+ when it is oxidated; this process of oxidation requires a first ionization energy: Na is supplied with energy in order to become Na+, so it is an endoergonic process; a strongly energetic one. When NaCl dissolves in water, the partially negative side of water molecule bonds weakly with Na+ (intermolecular bond, which has electrostatic character); in a qualitative view, forming a bond is an exoergonic process; since this type of bond is weak, the energy lost by Na+ isn't enough to bring Na+ back to Na.
Cl- has a lower energy than Cl has, so when Cl- forms a weak bond with the partially positive side of water molecule, its energy lowers and it's thermodinamically impossible for it to become Cl.
What you asked about poisoning power of Cl is a Biochemistry topic, but it's reasonable thinking that in a healty person's body at room conditions (e.g. not a scuba diver underwater) the total amount of Cl (Cl• radicals or Cl2 molecules) isn't large enough to cause health issues and it won't appreciably raise by drinking salted water. On the other hand, Cl- is part of our daily consumption of mineral salts.