Ideally, I would want to look at aqueous thermochemistry data for this, but all I can find references for is gas phase thermochemistry. However, if we run with that, we've got the following energy barriers to work with:
- HCl -92.31kJ/mol
- H2O -285.82kJ/mol
- HClO -74.48kJ/mol
I have included HClO because gaseous chlorine spontaneously reacts with water to form HCl and HClO.
(Just for reference, the heat of formation of HF is -272.55kJ/mol--getting two hydrogens out of that is almost twice as expensive as getting it out of water, so that ain't happening! Nobody's gonna be breathing fluorine here.)
So, how much energy does it take to get 2 hydrogens from various reactions? As a baseline, splitting water and releasing oxygen requires 285.82kJ/mol, plus metabolic inefficiency:
H2O + 285.82kJ/mol -> $\frac{1}{2}$O2 + H2
What about exchanging a chlorine for a hydrogen, converting water and HCl into hypochlorous acid and hydrogen?
H2O + HCl + 303.65kJ/mol -> HClO + H2
Looks like that won't happen, because it costs more! And directly splitting HCl?
2HCl + 184.62kJ/mol -> CL2 + H2
That turns out to look considerably cheaper than splitting water! So yeah, Chlorosians should produce chlorine gas! Except... chlorine reacts with water. Based on the data I have, the reaction
H2O + Cl2 -> HClO + HCl
should consume 119.04kJ/mol, which explains why producing hypochlorous acid directly looks disfavorable here. But in reality, that reaction actually does occur spontaneously, so I have to assume that there are solvation effects or something that make this analysis incorrect.
So, it kinda looks like Chlorosians would really produce hypochlorous acid rather than chlorine gas or oxygen. But... exposure to sunlight causes hypochlorous acid to dissociate into HCl and free oxygen. So, they end up getting to an oxygen atmosphere in the end after all!
EDIT: According to Chloride Oxidation as an Alternative to the Oxygen Evolution Reaction on HxWO3 Photoelectrodes producing chlorite from aqueous Cl- and H+ ions requires a slightly higher electrode potential than electrolysing water--so it seems that Chlorosian plants would be better of just using water as their hydrogen source, and Chlorosian animals will therefore definitely end up breathing oxygen.