What the Film - and John Carpenter - Say About It:
The film never addressed this issue, and Carpenter is notoriously taciturn about the questions everyone wants to ask, and extremely elusive in the few situations in which he has given people an opportunity to ask such questions, typically giving curt, dismissive replies like "I don't know" or even "I don't care".
On occasion, he is more willing to talk about the movie, and you can find a number of videos on YouTube featuring Q and A sessions following screenings of The Thing; there are at least a few with Carpenter himself, and many more with various members of the cast, which are usually quite interesting.
What the Short Story The Things Says About It:
However, there is a story that can provide us with some clues. It is not strictly canonical, and in fact, the author refers to it as "fan fiction", but he is being admirably humble to an extreme degree. The story is titled The Things, the author is Peter Watts, and it is written from the perspective of the Thing itself, just after the events of the film.
The story has won, or been nominated for, several major awards for science fiction, including the Hugo Award. It is short enough to read in a few minutes, and references to the Thing's ability to retain information are scattered throughout the text, but I can offer at least a few examples here.
The Thing seems to store its memories throughout its "body", for lack of a better word, because in the narrative, the Thing laments the fact that, as it was attacked and wounded by MacReady and his colleagues, it lost most of its intellectual capacity, including its memories.
It is important to mention that the story is so unique that it will be difficult to understand the text without reading the story in full. The Thing refers to the copies it makes of the people and animals it has assimilated as "offshoots", for example. It refers to assimilation as "communion". It also believes that assimilation is a desirable state of affairs, and is absolutely shocked to learn that people are so horrified by the prospect of being assimilated. The Thing is used to every life form in a given world being the same collective organism. This is the normal way of life for the Thing. It is therefore amazed by the fact that each life form on earth is a separate entity with its own mind, memory, and knowledge, which no other entity has access to. It is hurt and stunned by the violence with which the humans have reacted to its presence, and has never experienced such aggression in the countless ages since it came into being.
That said, here is a small sampling of the relevant portions of the text:
So much wisdom I had. So much experience. Now I cannot remember all the things I knew. I can only remember that I once knew them.
I remember the crash, though. It killed most of this offshoot outright, but a little crawled from the wreckage: a few trillion cells, a soul too weak to keep them in check. Mutinous biomass sloughed off despite my most desperate attempts to hold myself together: panic-stricken little clots of meat, instinctively growing whatever limbs they could remember and fleeing across the burning ice. By the time I'd regained control of what was left the fires had died and the cold was closing back in. I barely managed to grow enough antifreeze to keep my cells from bursting before the ice took me...
I'm used to finding intelligence everywhere, winding through every part of every offshoot. But there was nothing to grab onto in the mindless biomass of this world: just conduits, carrying orders and input. I took communion, when it wasn't offered; the skins I chose struggled and succumbed; my fibrils infiltrated the wet electricity of organic systems everywhere. I saw through eyes that weren't yet quite mine, commandeered motor nerves to move limbs still built of alien protein. I wore these skins as I've worn countless others, took the controls and left the assimilation of individual cells to follow at its own pace.
But I could only wear the body. I could find no memories to absorb, no experiences, no comprehension. Survival depended on blending in, and it was not enough to merely look like this world. I had to act like it—and for the first time in living memory I did not know how.
Even more frighteningly, I didn't have to. The skins I assimilated continued to move, all by themselves. They conversed and went about their appointed rounds. I could not understand it. I threaded further into limbs and viscera with each passing moment, alert for signs of the original owner. I could find no networks but mine.
Of course, it could have been much worse. I could have lost it all, been reduced to a few cells with nothing but instinct and their own plasticity to guide them. I would have grown back eventually—reattained sentience, taken communion and regenerated an intellect vast as a world—but I would have been an orphan, amnesiac, with no sense of who I was. At least I've been spared that: I emerged from the crash with my identity intact, the templates of a thousand worlds still resonant in my flesh. I've retained not just the brute desire to survive, but the conviction that survival is meaningful. I can still feel joy, should there be sufficient cause.
And yet, how much more there used to be.
The wisdom of so many other worlds, lost. All that remains are fuzzy abstracts, half-memories of theorems and philosophies far too vast to fit into such an impoverished network. I could assimilate all the biomass of this place, rebuild body and soul to a million times the capacity of what crashed here—but as long as I am trapped at the bottom of this well, denied communion with my greater self, I will never recover that knowledge.
I'm such a pitiful fragment of what I was. Each lost cell takes a little of my intellect with it, and I have grown so very small. Where once I thought, now I merely react. How much of this could have been avoided, if I had only salvaged a little more biomass from the wreckage? How many options am I not seeing because my soul simply isn't big enough to contain them?...
I remember losing myself after the crash. I know how it feels to degrade, tissues in revolt, the desperate efforts to reassert control as static from some misfiring organ jams the signal. To be a network seceding from itself, to know that each moment I am less than I was the moment before. To become nothing. To become legion...
And yet, not quite. I can barely remember—so much was destroyed, so much memory lost—but I think the networks recovered from my different skins stayed just a little out of synch, even reunited in the same soma. I glimpse a half-corrupted memory of dog erupting from the greater self, ravenous and traumatized and determined to retain its individuality. I remember rage and frustration, that this world had so corrupted me that I could barely fit together again. But it didn't matter. I was more than Blair and Copper and Dog, now. I was a giant with the shapes of worlds to choose from, more than a match for the last lone man who stood against me.
I highly recommend that you read the entire story - it is undoubtedly worth it, and if you're a fan of the movie, you'll find it quite interesting and entertaining. However, it does deviate from the movie in at least one regard: whereas, in the most popular view, both Childs and MacReady were still human at the end of the film (although Carpenter has always refused to make any serious remarks about this issue), the story presumes that Childs had been assimilated at some point during the movie.
What the Novelization Says About It:
We also get a bit more information about this question from the novelization, written by Alan Dean Foster. I quote from the FAQ section of the fan forum, Outpost 31:
Q: How smart is the Thing?
A: The Thing's level of intelligence is a function of its size. The larger the Thing, the more intelligent it is likely to be. The smaller the Thing, the less intelligent it will be. MacReady's blood test is directly dependent on this idea. The novel has Mac explaining his theory in greater detail than the film:
"When attacked, it looks like even a fragment of one of these things will try to survive as best it's able. Even a sample of its blood. Of course, there's no higher nervous system, no brain to suppress a natural instinct like that if it's in the best interests of the larger whole to do so. The cells have to act instinctively instead of intelligently. Protect themselves from freezing, say. Or from incineration. The kind that might be caused by a hot needle, for instance."
(Alan Dean Foster, The Thing, 169)
This perhaps also accounts for why the Norris spiderhead scurried from its hiding spot when it did. Maybe its body mass was not sufficiently large enough to form an intelligent brain center. Consequently, it didn't know enough not to blow its cover when the men still presented a danger.
On the other hand, a full-sized Thing is extremely intelligent. It is theorized that it has the combined intelligences of all the organisms it has ever assimilated. This is borne out by the fact that Blair-thing, having likely been a product of either the Norwegian dog directly or one of its descendants (Norris or Palmer), has the intelligence to build a non-terrestrial ship out of helicopter and tractor parts. Blair-thing "inherited" the intelligences of its previous organisms, the knowledge being passed into the newest assimilant.
Q: How is a Thing able to imitate the behavior of a person so perfectly?
A: This obviously has to do with how the Thing can retain the memories and brain patterns of its victims. The most detailed description of this is found in the novel as Blair explains to the other men how a Thing could take over a dog while maintaining its canine behavior:
'Blair's voice remained even, tutorial. "As you say, the body is only designed to keep so much organic material alive and functioning. Portion's of this dog's brain, for example, have been blocked off by new structures. The flow of oxygenated blood has been redirected."
'"In other words," Copper said quietly, "part of its brain has been turned off."
'Blair nodded. "Certain cerebral regions were dead before this animal died, having been supplanted in importance by new activity elsewhere."
'"What regions were kill ... were turned off?"
'"Difficult to say. There was massive parasitic invasion. Some of those which control portions of the memory, intelligence, and in particular individuality. Hard to tell with a dog, of course, be it dead or alive." He turned his gaze back to the interlocked bodies.'
'"One cell is enough. The DNA pattern of the new host is irrevocably altered. And so on and so on, each animal it takes over becoming a duplicate of the original thing."
'"You been into Childs's weed, Blair?" Norris muttered.
'Blair's fist slammed onto the table. "Look, I know it's hard to accept! I know it's difficult to picture an enemy you can't see. But if that stuff gets into you system, in about an hour --"
'"It takes you over," Fuchs finished for him.
'"It's more than that, more than you becoming a part of it. The 'you' is gone, wiped out, shunted aside permanently by a new set of cellular instructions. It retains only what it needs of the original, the way it used the memory patterns of the Norwegian dog to make certain it acted in a recognizably doglike manner."
'"It licked my hand," Norris murmured, "as it was being chased by those guys in the helicopter. It came right up to me and licked my hand and whined for help."
'Blair nodded. "Sure it did. It keeps anything useful. This organism is highly efficient, not wasteful. And it's clever. Much too clever for my liking."'
(Alan Dean Foster, The Thing, 81, 82-83)