The leaves contain high levels of Oxalic acid, but the oxalic acid content of the tubers depends on the variety. There are two groups of cultivars: those with sour and those with sweet tubers. The tubers of the sweet cultivars reportedly contain no more soluble oxalates than spinach, beet leaves, chard, etc, and this is contained mostly in the skin. Leaving them in the sun further reduces the oxalic acid content and sweetens them. They can then be eaten raw, or cooking will further reduce oxalic acid levels.
The sour tuber cultivars do indeed require processing: they're soaked for a month, then left in hot sun and freezing nights until entirely dehydrated, at which point they are a storable crop called khaya in Quechua.
We grow a sweet variety and they're really great! They're a pretty happy hands-off crop, and even work well in a no-dig garden since the tubers are all just below the surface and can be harvested with minimal soil disruption. It's very easy to miss some tubers when harvesting, so if you have mild winters they'll all spring up again in the same spot the following year.
I've mostly just summarized the article at Practical Plants, but you can take a look here for more info:
http://practicalplants.org/wiki/Oxalis_tuberosa
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Written from Xardín do Cernunnos, a little valley permaculture homestead in the beautiful hills of Galicia.
