The first year we had a garden we planted, among other things, some collard greens. I don't even remember the variety, since this was back in the days before any sort of record keeping; although I'm pretty sure the seed packet is filed away somewhere, still more than half full (I keep old seeds around and try to start them every once and a while on an experimental basis—but that's a topic for another day). Anyways, those collards grew well enough, and, as collards do, they went to seed; and in the way of things many of those seeds were scattered all over the garden. The next spring, well before it was time to sow seeds outside, I noticed dozens of little collard plants springing up all over, and happily welcomed their presence. Here was the garden well-started, and we didn't even have to do any work!
My gardening style is pretty relaxed in many ways. I have a general vision for what I want to get done, but if circumstances turn me in another direction I'm not going to make extra work for myself to bend things to my will. And that means that if I notice a vegetable—or anything interesting-looking and not an obvious weed—sprouting out of place in the garden I'll tend to let it go for a while and see what it turns into. Basically, I love plants and don't want to kill them unless I absolutely have to.
Some of the time that works out pretty well. The collards are a great example: we enjoyed the offspring of that first planting for many years (though their run seems, sadly, to be nearly at an end now). Last year's best crop might have been the butternut squash that sprouted from seeds in the compost, which we let grow because it was in amongst rows of root vegetables decimated by rabbits and left to be swallowed by weeds. The vines tangled all over the adjacent few rows and made walking difficult in spots from time to time, but we got over a dozen good-sized squashes for no cost and with almost no labor.
Other instances of mercy, though, didn't turn out to well. The self-seeded Brandywine tomato seemed like a blessing when I noticed it not too long after the two Brandywine's we had planted were killed by cutworms (which happened just after I'd given away the rest of the seedlings), but it was ill-placed and started too late and didn't manage to produce a single edible fruit before it was felled by late blight towards the end of the season. Several volunteer cherry tomato plants have grown much better but never ended up producing anything other than small sour fruit and a great deal of inconvenience as they sprawled over paths and got in the way of other, deliberately planted, crops.
Still and all, we're inclined give self-seeded plants a shot. After all, they've already beaten the odds to reproduce themselves: sometimes even the seeds we plant on purpose in carefully manicured seed-beds don't manage to germinate. I did learn though, that whenever possible we should transplant volunteer seedlings to convenient spots. Being able to harvest abundant crops of delicious collards is even better when the plants in question aren't perched right on the edge of a bed, severely impacting navigation in the garden—or, even worse, off in the side yard or something (birds do remarkable work) where they don't get cared for or harvested. But when they're in a place where they belong it's like gardening magic, and I smile every time I see all that free food growing.