I like this question!
TL;DR: Mr Filch and Mrs Norris communicate through verbal and physical cues and are aided by a "mysterious power" (Chamber of Secrets - chapter 8, The Deathday Party) that heightens the connection between them. There is no definitive canon answer to this question.
Now for the long answer.
Mrs. Norris is a regular cat; she does not have any magical powers.
Filch owned a cat called Mrs Norris, a scrawny, dust-coloured creature with bulging, lamp-like eyes just like Filch’s. She patrolled the corridors alone. Break a rule in front of her, put just one toe out of line, and she’d whisk off for Filch, who’d appear, wheezing, two seconds later.
Philosopher's Stone -- page 99 -- Bloomsbury -- chapter 9, The Potions Master
That said, canon indicates that Mr Filch and Mrs Norris do share a symbiotic connection of sorts, fueled by some kind of unknown force, that goes above and beyond the normal pet/owner relationship.
Harry, [backed] away from the accusing stare of Mrs Norris, but not quickly enough. Drawn to the spot by the mysterious power that seemed to connect him with his foul cat, Argus Filch burst suddenly through a tapestry to Harry’s right, wheezing and looking wildly about for the rule-breaker.
Chamber of Secrets -- page 96 -- Bloomsbury -- chapter 8, The Deathday Party
Mr Filch actually cannot see Dementors and this was confirmed at J.K. Rowling's website in regards to a question about Arabella Figg and her claim to the Wizengamot that she saw the Dementors that attacked Harry and Dudley in Order of the Phoenix; she was twisting the truth for Harry's benefit. Unfortunately, J.K. Rowling has taken down all her extra tidbits regarding Harry Potter from her website so she can move it over to Pottermore. Judging by how slowly Pottermore has moved so far, who knows how long that will take to accomplish -- years? But, no, Squibs cannot see dementors. Canon doesn't mention Filch and Thestrals, so we actually don't know one way or the other whether he can see them or not (we don't know if he's seen death either), but we do know Filch can see Peeves and the ghosts.
Animal/human communication is a complicated thing. Animals learn cues from people and vice versa. Mrs Norris clearly has more heightened senses than Mr Filch does. For example, Mrs Norris comes across Harry, Ron, Neville, and Hermione under the Invisibility Cloak in Philosopher's Stone (chapter 9, The Midnight Duel) and she stands there for a long moment, clearly sensing the presence of something, but she cannot see through the Invisibility Cloak. She returns to Filch and somehow communicates to him that there are people present and out of bed after hours.
How does she communicate? As you mention, a specific sound or meow is a good possibility. It could be that she's been trained to paw the ground a certain number of times, depending on the infraction: Students out of bed -- one paw; Peeves destroying Hogwarts' property -- two paws; general suspicious behavior, such as Nearly Headless Nick's Deathday party -- three paws. Many service animals are trained to communicate with their owner using specific body movements, such as pawing or laying the head on the owner's lap or sitting automatically when the owner approaches a street corner.
As for Mr Filch and Mrs Norris sharing a special language, I don't think that's likely and here's why: Parseltongue is a magical language, passed down only by Salazar Slytherin to his descendants. First, Mr Filch is a Squib and has no magical abilities -- I don't think he would be capable of understanding or speaking a magical cat language. Second, canon only indicates Parseltongue as a language shared between humans and reptiles; no other human-animal languages are demonstrated (although Dumbledore clearly has a strong channel of communication with Fawkes; no actual language is revealed, though)
A good guess, I believe, is that Filch uses verbal commands (as we see him do in the quote you provided from Philosopher's Stone and Mrs Norris uses physical cues to communicate with each other. Canon states there is a mysterious power at play between Mr Filch and Mrs Norris, so there is an aspect of their communication that cannot be explained by canon or anecdotes.