I'd let them grow. I turned in some bidwell casaba melons to my compost pile last year. This year I ordered more seeds and lost them. No worries, I have some growing in the compost pile which I am going to transplant into a bed I made for them yesterday.

I had some tomatoes come up in a small compost pile spot -- I got a lot of good tomatoes from them!

Leisa

I planted a pumpkin plant and a cantaloupe last year out of my compost. Both did really well! As I always say in the garden, "give it a try! Worst case scenario, it doesn't work." I would plant seedlings sooner than later, they seem to come out of the pile and transfer with more roots when you do it early.

One year my g'kids planted pumpkins in the cattle paddock area. Cinderella could've ridden to the ball in them.

If you can, let them grow in the pile. Thin of course, but the fertility and moisture of a cooling or finished heap is a cucurbit's ultimate fantasy!

If you suffered any late blight last year, you'll want to get rid of any potato volunteers and their foliage (bag them and set them out with the trash). It can overwinter in last year's potatoes. Or at least that's the advice I was given at a talk I went to.

I would think that if the compost pile were properly "cooking" that the internal temp of the pile would not support plant growth. If you have plants growing, stir up the pile and let them compost into it too.

I have good success with tomatoes, spinach, Swiss chard and several flowers that pop up after I use compost into garden.

If they're from a hybrid plant, they won't taste the same, but...as long as they do well, I'd give 'em a whirl...transplanted.

I get a lot of compost volunteers too. Our compost is wonderful, so i am not sure why the heat does not kill the seeds. In any case, it doesn't, but since as i said the compost looks, smells, feels so good, we do not worry about it. Yes, we have heaps of worms in the compost, and the effect of adding compost to anything we grow is pretty clear (good!)

What I do with volunteers depends on what I get. If they are squash, they will be mongrels (either f1, f2, or some combination of C pepos). If they are not C. pepo, then they are C. moschata, we only grew butternut last year, and our best specimens were compost pile volunteers! Every time we have allowed them to grow and fruit, we have been very impressed with the disease resistance, vigor, production, and flavor of the fruit. So we keep all the squash. If they are cantaloupe melons, we also keep them because we have been impressed with their flavor in the past. I would yank out and toss the watermelons though, simply because any seed would be from last year variety, which i do not want to grow again (watermelon is in its own species so it will not cross unless you plant different watermelon varieties).

If I get potato seedlings, I transfer them to my 'impure' potato patch. I have two patches, in one goes everything (potatoes that sprout from the store or the compost) in the other, only certified seed potatoes, and a lot of care is taken to not 'contaminate' that patch. Needless to say, the potato patch with 'everything' does a lot better and produces like crazy. Go figure.

If they are tomatoes I yank them out, break them, and toss them back in the compost heap. Volunteer tomatoes we have grown in the past really sucked. Your experience might differ, and we did try it and grew them once. Never again for us.

I also get volunteer red amaranth, which I allow to live and feed me salads and greens in the summer, while delighting us with the red splashes all over the garden.

Onions we keep too, plant them near the roses. Most of my onions are sprouted from the grocery store, ditto for the garlic. I do plant some starts and seeds as well.

If you know what they are, perhaps they are better than those you start from seeds.
By the law of naturall selection, not all seeds survive and germinate
and thosr that do, are the best and strongest that have survived the harsh elements.