I don't know about the elephant garlic seeding itself, but I do have the Allium giganteum plants, which occasionally appear to make seeds but have never seeded themselves (they've only multiplied by bulb division).
As far as chives, they definitely will self-sow pretty exuberantly... but the flowers are very attractive to bees, and you can collect the seeds and use them to grow sprouts to put in salads!
