What are the implications, if any, of brewing beer with water treated by a water softener? I've read lots of conflicting information and am looking for something more scientific and definitive.
A concern with using softened water, whether all grain or extract, is that the large amount of sodium introduced will give a salty taste to the beer.
On the scientific perspective, Mr Wizard, at BYO writes
If you can get a city water report or have your water tested, then that may help you decide if you will end up with too much sodium or not after passing it through the softener.
Assuming the water is fairly hard (otherwise it wouldn't need softening), it's better to skip the softener and use the hard water as is. This is fine if you're brewing extract, since the concentrations of hardness causing minerals are not so important, although if your bicarbonate or carbonate levels are over 300ppm then you might consider diluting, although taste the water first after boiling it and see.
If you're all grain brewing, then you'll need the water report, and use that as a basis to determine how much to dilute the water to get the dissolved minerals to acceptable levels. You dilute the water with distilled/deionized/RO water to reduce the overall amount of dissolved solids in the brewing liquor.
With all grain, it's really best to skip the softener, otherwise you'll have the counter-productive arrangement where calcium and magnesium are first removed by the softener and replaced by twice as much sodium, only for the brewer to then add back in more calcium and magnesium, since these are needed in the mash. By diluting your hard water you can end up with a workable water profile without introducing all the unwanted sodium.
Two considerations not mentioned thus far: 1. pH is important for mashing all-grain. If your pH is low (either with your softened water or with your tap water), you may experience problems. To the best of my knowledge, it is high or low pH that causes all-grain brewers to add minerals or RO water.
I'd suggest you should check your pH using some inexpensive strips. My local homebrew store had these for $4. Alkalinity can be checked by a pool store or with a pool test kit. The pH of your soft and hard water will probably determine your strategy more than anything else.
You can find a guide to each mineral on this blog page by Beersmith. It has a lot more information on each mineral than in my post or mdma's. Note one concern regarding carbonate (e.g. calcium carbonate) is the effect on pH.
From his page: