Before answering this question, it is instructive to provide some background
on the principles of human transfiguration.
Contrary to Himarm's answer, the assumption in the question about human transfiguration is correct - transfiguring oneself into an animal would result in the loss of human mental faculties, including the capacity to perform magic or, in Krum's case, to pursue the goal of saving Hermione. This is stated in Dumbledore's comments on “Babbitty Rabbitty and her Cackling Stump” in the Tales of Beedle the Bard:
Animagi do not retain the power of human speech while
in their animal form, although they keep all their
human thinking and reasoning powers. This, as
every schoolchild knows, is the fundamental
difference between being an Animagus, and
Transfiguring oneself into an animal. In the case
of the latter, one would become the animal
entirely, with the consequence that one would
know no magic, be unaware that one had ever
been a wizard, and would need somebody else to
Transfigure one back to one’s original form.
With this in mind, one possible answer to the question is that Krum's transformation into a shark was a half-successful attempt at an Animagus transformation. However, there are numerous pieces of evidence to rule out this possibility. Had Krum indeed been attempting an Animagus transformation, he would have had to register his efforts with the Ministry of Magic, who closely monitor actual and aspiring Animagi due to the dangers involved in the transformation. However, there is no indication that Krum underwent any such registration process - we know that Hermione was already familiar with the registry well before the second task, and she would likely have consulted it again before threatening to out Rita Skeeter as an unregistered Animagus (which Hermione did well after the second task) to make sure that Skeeter had not registered since Hermione had last checked the registry.
In so doing, Hermione would doubtless have encountered information about Krum's attempts to become an Animagus if any such information did indeed exist. The fact that we never hear any mention from Hermione about Krum attempting to become an Animagus therefore indicates that he never made any such attempt.
Moreover, we know that Krum would not have had much time to prepare for the second task - the champions received their clues about the content of the second task on 24 November 1994 and the second task was on 24 February 1995. So even if we assume that Krum miraculously resolved his clue in no time at all, he would have a maximum of 3 months preparation time. In contrast, James Potter, Sirius Black and Peter Pettigrew took almost 3 years to become full-fledged Animagi, making it extremely unlikely that Krum would have even entertained the possibility of mastering such a complex piece of transfiguration in the short amount of time he had available. Indeed, part of the process of becoming an Animagus involves drinking a potion that requires two consecutive, cloudless full moons to mature. There were only three full moons between 24 November 1994 and 24 February 1995, and the chances of two of these being cloudless and consecutive would have been very low given that they all fell in winter months.
Finally, Animagi in animal form tend to have visual characteristics that are reminiscent of their human forms (e.g. the rings around Minverva McGonagal's square spectacles matched the markings around her eyes in cat form; Rita Skeeter's glasses corresponded to the markings around her antenna when in beetle form), whereas there is no mention of any visual features on the shark part of the half-transformed Krum that correspond to his human appearance. Thus, we can safely rule out the possibility that Krum attempted an Animagus transformation in the second task, forcing us to concede that his half-shark, half-human form was the result of non-Animagus-like self-transfiguration and necessitating an alternative answer to DVK-on-Ahch-To's question.
Given the above-quoted comments from Dumbledore's commentary on “Babbitty Rabbitty and her Cackling Stump”, it seems incontrovertible that if Krum had fully transfigured himself into a shark, he would indeed have lost all human mental faculties and magical powers. It therefore stands to reason that the only thing that could have enabled him to attain his human mind was the incompleteness of his transformation. However, this raises several other questions, some of which are already mentioned in existing answers to the original question:
Since Krum's head was transfigured into the head of a shark, it follows that his brain also transformed into a shark brain. How would the non-brain part of his body enable him to keep his human mind?
Ludo Bagman's comments on the second task, as cited in Slytherincess' answer, imply that the incompleteness of his transfiguration reflected a deficiency in his spell work:
‘Mr Viktor Krum used an incomplete form of Transfiguration, which was nevertheless
effective, and was second to return with his hostage. We award him forty points.’
This indicates that the transformation was effective despite being incomplete, hence the incompleteness counted against Krum.
But if the incompleteness was necessary for him to retain his mental faculties,
why was it judged to be a deficiency?
- If complete transfiguration into an animal results in the loss of human mental faculties, how could Draco Malfoy remember being transformed into a Ferret by the imposter Mad-eye Moody (really Bartemius Crouch Junior).
I think the answer to 1. is that in the wizarding world, mental faculties do not purely reside in the brain. For example, Tom Riddle's diary - and to a lesser extent some other Horcruxes - clearly demonstrated that it held Voldemort's thoughts and memories and that it could apply his mental abilities and motivations within its own immediate context. The Horcruxes did not contain anything resembling a human brain, leading us to conclude that brains are not essential for generating thought and memory in the Harry Potter universe. Similarly, we see that Harry gained certain mental capabilities - such as the ability to communicate with snakes - when a piece of Voldemort's soul latched on to him, again demonstrating that cognitive functions can derive from the soul, and not the brain. When dementors suck out a person's soul, they suck it through their mouth, indicating that it resides somewhere in the region of the lungs and not the brain - if the soul was 'in' the brain, presumably they would suck through an orifice more closely connected to it, like the ears. Indeed, when Voldemort had no body - and thus no brain - he was still perfectly capable of thought and memory. Hence, we see that the soul can underpin memory and thought and that it does not reside in the brain, which explains how Krum could retain human memories and thought processes despite having a shark brain. This of course leaves open the question of why a non-Animagus human who is fully transformed into an animal would not likewise retain their mental abilities (they don't die, so their soul must still be around somewhere), but I think that would be beyond the scope of the original question.
I think the answer to 2. is that Bagman was probably somewhat imprecise with his language. The deficiency in Krum's transfiguration was probably not its incompleteness, but the fact that it was not optimally suited to his goal in the second task. My guess would be that Krum's intended transfiguration differed from the actual result in that Krum probably intended to give himself a shark's gills and shark fins, and perhaps the aerodynamic skin of a shark, but to retain enough human form to preserve his mental faculties. The deficiency in his transfiguration, then, was that he did not end up with fins and smooth shark skin, and he had a more shark-like head than he had wanted. Bagman described this failure to fully achieve his intended transformation as 'incomplete' simply for lack of a better word.
And I think the answer to 3. is that there is no clear evidence that Draco Malfoy did have any direct memory of his experience as a ferret following the imposter Moody's transfiguration. There are incidents that imply that he had some awareness of what had happened during his brief time as a ferret, for example:
Hagrid's smile faded off his face.
"Yeh'll do wha' yer told," he growled, "or I'll be takin' a leaf outta Professor
Moody's book. . . . I hear yeh made a good ferret, Malfoy."
The Gryffindors roared with laughter. Malfoy flushed with anger, but apparently
the memory of Moody's punishment was still sufficiently painful to stop him from
retorting.
However, Malfoy's reaction here could simply reflect second hand knowledge of what Moody had done to him, along with his memory of the immediate aftermath when he was transfigured back to human form. He would have remembered transforming back into a human with pain in his body after being bounced on the floor as a ferret, and Crabbe, Goyle and his other associates would presumably have told him about the incident upon realising that he did not remember it fully. So, in light of Dumbledore's comments on Babbity Rabbity, it seems likely that Malfoy did not remember what happened to him when he was a ferret, but that he had subsequently learned about it from his experience in the immediate aftermath coupled with accounts provided by others.
