TallRider

I'm actually pretty sure that it doesn't work both ways - or at least not both ways equally. In my understanding, a worn chain can wear out new cogs, but worn cogs won't wear out a new chain. At the very least there's less danger of worn cogs or chainrings wearing out a new chain. Here's why:
* A worn chain is elongated. This means that most of the force exerted by chain on cogs or by chainrings on the chain is transferred through one or a few gear teeth. This causes the teeth to wear much more quickly than if the force was transmitted along a larger number of teeth. (It's also the same reason that 11t cogs wear out quickly if you use them regularly - even a new chain isn't in contact with very many teeth. Fortunately most riders who have an 11t cog don't use it much.)
* A worn cog or chainring has its teeth in a shark-fin shape. However, the teeth are still the same distance apart as they were before, and so force on the chain is relatively equally-distributed on the chain links even from worn teeth. Moreover, even if force isn't as equally distributed among chain links, the chain isn't directly disadvantaged by this to the degree that chainring teeth are.

For very-worn cogs or chainring teeth, there is some danger of wearing out a new chain. This is because more shark-finned teeth contact the chain roller at an angle that forces the chain up and away from the center of the cog or chainring's rotational axis (again, this is more pronounced on smaller cogs/chainrings with fewer teeth, because there is more curvature per tooth). The further away the chain is pushed from the cog or chainring's axis of rotation (and for a given amount of tooth wear, this is inversely proportional to the number of teeth in the chainring or cog because more teeth equals less curvature per tooth) the further the distance between the teeth. So worn cogs are more of a danger to wear out new chains than are worn chainrings.
This forcing-the-chain-up usually causes the chain to skip. One reason why, practically-speaking, you should be able to ride a cog without worrying how it's going to affect your chain, so long as the chain doesn't skip.