The bushing (or it's equivalent formed in the inner plates) wears the leading 'side' of the chain pin. You can see it rather easily on a not-terribly-worn chain as one/two--depending on the bushing design--extremely polished areas.
On badly worn chains, the polished area will be straight grooves, the integrated bushings may be badly worn to the point that the rollers are rattling, and the side plates are worn to the point that the chain can be flexed sideways into a rather tight circle compared to a new one. Such a chain, even if it doesn't skip, tends to poor shifting performance, since links would rather flex sideways than follow/pull on i their neighbors.