Well, when it comes to tribble ownership, I guess I'm ahead of many others...
Supposedly, fan visitors to the set of TOS, in the weeks following the episode being filmed, were given a tribble as a souvenir, and the fuzzy little props were also often sent to conventions for charity auctions throughout the 70s, 80s and 90s. There were tribbles popping up all over Desilu for months. No one thought to create certificates for them at the time, so it can be hard to prove a supposedly screen-used tribble as truly genuine. Writer David Gerrold took home lots of them, and they lived in his garage, slowly dwindling down over the years. The genuine tan ones easier to identify, as the director was apparently unhappy with the pale colour and ordered the fur to be dyed darker with toothbrushes, giving a unique two-toned effect not seen on commercial, fake-fur, acrylic fabric. It's much harder to identify the small white TOS ones, since that acrylic fur is "off the rack" and easily found in fabric stores.
The ones sold by the Roddenberry's Lincoln Enterprises mail order firm were commissioned out to cottage industry people and these were facsimiles, and none were overdyed. Many had squeakers sewn inside. The catalogs did not claim that they were genuine, screen-used props.
I helped David Gerrold auction off a small, genuine, screen-used tan tribble, to which he'd attached a small pet collar, a lead, and a yellow plastic "Tribble on Board" sign. At that New Zealand convention, he discussed how to identify the genuine tan ones, and how there are many replicas which people attempt to sell as originals - hence he'd started doing letters of provenance when he sold them for charity fundraisers.
Although that baby tribble went for a tidy fortune (for charity) and I missed out, I do now have a 1967 TOS tan tribble: supposedly the very last one of its type, found in the back of a drawer when David Gerrold and his son (aka "The Martian Child") decided to clear out their garage. David wrote a wonderfully-worded certificate for it and sold it on eBay, along with an autographed first edition copy of his "Trouble with Tribbles: The Making of the TV Episode" book, and newly-printed, signed, replica scripts of his TOS and TAS tribble episodes. A friend of mine bought the package deal, then regretted the many US dollars he'd spent, so I bought it off him.
From the certificate: "Over 500 prop tribbles were made for the show, and reused throughout. This tribble is one of the actual tribbles used in the episode. It was given to me by one of the prop men for the series after the show was completed."
By the way, when the package arrived, I quickly emailed my friend to admonish him that he shouldn't have packed any quadrotriticale inside for the tribble to eat because, when I opened the box, it was now filled with tribbles - and I wasn't sure which one was the original! ;)