It is shown in the first book that a lot of wizards live in the muggle world, with day jobs and muggle clothing to keep themselve discreet. They even forget to swtich into muggle clothing and celebrate openly in their robes, showing that wizards are common enough in the muggle world that people might take notice on a particular day when everyone is excited and not bothering with complicated precautions of subterfuge.
So clearly there are plenty of wizards in the world, running ordinary day jobs, who could guarantee that new witches and wizards have no trouble going to Hogwarts, especially since, as described in the books, it has all the makings of a very typical secondary school, if you ignore that everythign they learn about is magic instead of science and math et cetera.
And, let's not downplay the importance of helpful mudblood parents. Hermione's parents in particular seem very supportive of her being a witch, and they'd likely go along with anything that would make this easier for her. We can assume, with a few exceptions (Harry in particular) that most parents wouldn't even bother to report anythign unusual, and that would probably be enough. It's not as if the students are disappearing - they'd come home every summer and they'd be able to spend time with their Primary School friends. So if they're clearly coming and going to a new Secondary school, and then coming home in the Summer and having no problems at all, why would anyone report anything unusual?
Not to mention, there are always memory charms to deal with particularly difficult cases.