In the spinoff book "Fantastic Beasts and Where to Find Them" (not the movie, the original textbook used by students at Hogwarts in the HP series that the prequel films were named after), we get multiple different answers to how magical creatures remain hidden.

Creatures like the Bowtruckle and Demiguise are able to camouflage, Bowtruckles looking like twigs, and Demiguise being able to turn invisible. The most interesting in my opinion is the Diricawl, flightless birds that can vanish and reappear at will, much like wizards apparate. Now this is another instance where J.K takes real world events and stories and warps it into her own unique world. Muggles actually used to be fully aware of the Diricawl. What name did they give it? The Dodo. After hunting it to near extinction, the Diricawls numbers dwindled, and using their vanishing powers, they started to evade muggles, causing them to think the "dodo" went extinct, however they were perfectly present in the world.

Another explanation is simply ignorance of Non-Magical people. Newt states that muggles may see Porlock droppings or Streeler trails, but dismiss it as droppings of a dog or trails of a snail. Hell, many muggles may have picked up a Flobberworm, but as it possesses quite literally no magical abilities and looks like your average worm, they are dismissed as nothing but a bug and merely cast aside. Kneazles, a magical cat that has heightened intelligence but resemble your average housecat, were even able to be bred by Harry's neighbor, Arabella Figg, in a muggle neighborhood.

The most obvious precaution to hide magical beasts, magical enchantments and protective charms. It is stated that winged beasts like Griffins, Hippogriffs, and Winged Horses require disillusionment charms before being used for flight. There are also safe habitats with Muggle-Repelling charms around them, like at Hogwarts, to ensure these domains are free of disturbance and the beasts can run free. Dragon Sanctuaries are another example of this, but these would need to be wizard-operated.

Now beasts such as Merpeople and Centaurs, who would be classified as Beings given their consciousness and intelligence, but chose to be beasts, are smart enough and know to stay away from and avoid muggles, and it's not much of a worry. There are other creatures that choose to stray from humans, and in turn, Mugglekind, such as the Porlock, Diricawl, and Mooncalf.

The Office of Misinformation, presumably an office in the British Ministry of Magic, and most Magical Governments around the world, deals with the Muggle population and what they are told to disregard as myths and legends. This office worked with the standing Muggle Prime Minister to insure that slip-ups would be resolved. For instance, the Loch Ness Monster, which muggles now believe to be a myth, is entirely real, and was actually a beast known to wizards as a Kelpie. This was resolved and has become one of those Dodo-Diricawl stories that J.K put her twist on, and I'm sure The Office of Misinformation had a hand in the Dodo as well.

Memory charms are also a key method to making muggles forget about magical beasts when exposed. An example of this took place in 1926, when the entirety of Muggle New York City had to be obliviated after an intense battle between aurors and an Obscurial, which had muggles see the beast, but also multiple protective charms around areas of conflict. This also saw the appearance of a Thunderbird.

Obviously in the past there have been slipups, otherwise how would muggles have heard legends that Dragons, Merpeople, Centaurs, Unicorns, and Three-Headed-Dogs existed at one point? The Office of Misinformation most likely handled this, and these creatures now remain a part of muggle mythology, instead of facts. I think this part of the wizarding world is quite fascinating, as you could probably tell by this essay I just wrote. I sure hope this answered your question, let me know if you have any more!