[Maerbenn]

From Myths Transformed:

It seems clearly implied in The Lord of the Rings that trolls existed in their own right, but were 'tinkered' with by Melkor.

From The Lord of the Rings, Appendix F:

In their beginning far back in the twilight of the Elder Days, these were creatures of dull and lumpish nature and had no more language than beasts. But Sauron had made use of them, teaching them what little they could learn, and increasing their wits with wickedness.

From The Letters of J.R.R. Tolkien, Letter #153:

I am not sure about Trolls. I think they are mere 'counterfeits', and hence (though here I am of course only using elements of old barbarous mythmaking that had no 'aware' metaphysic) they return to mere stone images when not in the dark. But there are other sorts of Trolls beside these rather ridiculous, if brutal, Stone-trolls, for which other origins are suggested.

So it seems that the Stone Trolls in The Hobbit were not creatures tinkered with (bred with Men) by Melkor or Sauron; they were of the original stock of Trolls. The other Trolls (of Melkor and Sauron) were corruptions of this Troll stock (they could withstand the Sun). That one of the Trolls in The Hobbit (William) has a feeling of pity for Bilbo seems to support this.

But; I'm aware that I am on shaky ground citing William's "pity", because JRRT in that same letter says:

I do not say William felt pity - a word to me of moral and imaginative worth: it is the Pity of Bilbo and later Frodo that ultimately allows the Quest to be achieved ... There is no more 'pity' here than in a beast of prey yawning, or lazily patting a creature it could eat, but does not want to, since it is not hungry.