Why don't Muggles notice magical creatures?
Most of the Magical Creatures of the Wizarding World have their own form of hiding from potential danger and from being noticed by the Muggle World. Thus, some beasts and beings need wizarding assistance from the Ministry of Magic.
The Department for the Regulation and Control of Magical Creatures is the second largest in the Ministry of Magic, and it is this department that deals with the varying needs of magical beasts and beings.
Safe Habitats
Many Muggle-Repelling charms protect beats and beings from being discovered by trespassers.
In different parts of the world, either only small areas or whole and complete areas have been made unplottable to Muggle eyesight and Muggle technology.
Not always do these safe habitats work, for not all creatures like staying within them, such as dragons and the pogrebin. In these cases, wizarding supervision is required to ensure these creatures remain hidden to the Muggles.
Controls on Selling and Breeding
The Department for the Regulation and Control of Magical Creatures has set several rules and restrictions on the selling and breeding of magical creatures, mostly dragons.
It has been made illegal to create new magical beings by breeding among different types of creatures.
There are fines imposed upon the breeding and selling of magical creatures, and big penalties apply for those that do not follow them.
Disillusionment Charms
Restrictions and rules also apply to wizards and witches that have desired to have some of these magical creatures as beasts.
Those that possess several creatures, such as a Hyppogriff, are bound by law to perform disillisionment charms daily on their creatures so that the Muggles can not perceive them as they are.
Memory Charms
If a Muggle does notice a Magical Creature, memory charms are set upon them so they can either think they saw something else, or forget the whole episode.
The memory charms are either performed by the owner of the beast in question, or by an Obliviator sent from the Ministry of Magic.
The Office of Misinformation
Involved only in urgent and severe Muggle-Magical collisions.
Liasons sent from this department will work with Prime Ministers of the world in order to try and persuade them of non-magical reasons as to why big occurrences have happened, including sights of the Loch Ness Monster.
Nothing is ever really revealed.