The tannic acids in peat slightly reduce pH, and are anti-bacterial, but as far as softening water significantly, other than the acids reacting and neutralizing a little calcium, reducing the strength of other minerals is a bit of a stretch.
I have used peat, and fallen leaves and had slight success with my hardish water (normally @ 250 ppm, alkalinity @100ppm, and normal pH 7.8 or higher,) but the tannins only dropped water a few 10ths in pH, no reduction in the Total Hardness, but reducing buffering capacity from 100, to about 60.
Soft water fish reacted well to the tannins over time, so for those species I soaked lots of leaves in fall, and put peat in filters or mesh under influent pipes, and also added rain water.