[Angry Hill Troll]

Gandalf not telling anyone about Thráin's Ring is a tricky one. When did he actually know for sure that the Ring had been taken from him? I can't recall, but I don't think Thráin mentioned it at the time, and that it was a later guess of Gandalf's.

Actually Thrain did speak of the Ring, but did not tell Gandalf who he was (The Quest of Erebor in UT)

...I had entered Dol Guldor in disguise, and had found there an unhappy dwarf lying in the pits. I had no idea who he was. He had a map that had belonged to Durin's folk in Moria, and a key that seemed to go with it, though he was too far gone to explain it. And he said that he had posessed a great Ring.
"Nearly all his ravings were of that. The last of the Seven he said over and over again...
"Now [meeting Thorin some ninety years later] I remembered it all again, and it seemed clear that I had heard the last words of Thráin the Second, though he did not name himself or his son; and Thorin, of course, did not know what had become of his father, nor did he even mention 'the last of the Seven Rings.'

It's not clear that Gandalf ever spoke to Thorin about Thráin's ring. The Quest of Erebor indicates the two had quite a strained relationship, and it was only with great reluctance that Thorin heeded Gandalf's admonition to send Bilbo with them. Since Gandalf was planning an attack on Dol Guldor at the same time, the last thing he would have wanted was for Thorin to forget all about retaking Erebor and instead trying to get his father's ring back.

Would Thorin have even known that his father had a Ring without Gandalf telling him? hmmmm...