Chives are usually propagated by divisions. However, they can be grown from seed with a little extra planning time. So if you don’t have a neighbor with chives to divide and give away, try starting your own chives from seed. Once you have them up and growing, they’ll readily divide and bunch thickly in your garden.
They’ll also self-seed every year, starting the second year when they produce flowers, giving you more and more chives every year (unless you cut the flowers before they produce seed).
There are supposedly several different varieties of onion chives in the USDA seed collections, but I’ve only ever seen one variety offered by any seed catalog ever. Perhaps there are different varieties among these (they all look the same… but you never know), but if there are, the seed companies themselves don’t distinguish any of them by any particular name.
Both garlic chives and onion chives are outbreeding plants – the flowers won’t accept pollen from itself. It must be pollinated by another flower.
Onion chives will not cross with garlic chives or any other allium species. And since we really don’t have access to (or know we have access to) different varieties of onion chives, there’s really no need to bag the flowers of any onion chives you grow. Just let the pollinators do their work, then collect the seed heads before they drop all their seeds. Let the flowers finish drying out in a warm, dry area, well out of the sunlight. Shake the seeds from the flowers and winnow with a fan on the lowest setting if necessary.
Same goes for garlic chives. I think a paragraph in Seed to Seed by Suzanne Ashworth says it best:
If different varieties exist and were grown near each other, they would be insect cross-pollinated. ☺
So just collect the flower heads before they drop all their seeds, and do the same as you would for onion chives.