Adopting a "found document," first-person stance goes way back. Both Frankenstein and the original Blacula — Dracula — are written as first-person frames consisting of excerpts from letters and journals — narrative formats with which the tubercular, bedridden readers of the era were already familiar. For them, these layers of voyeurism and peeping-Tomism likely added to the frisson of horror and sexy, sexy danger evoked, respectively, by Shelley and Stoker.
But for all the emotional immediacy, a first-person cameraman/narrator is also an inherently delimiting and story-circumscribing device. Like the first-person narrator in written fiction, she's privy to only the circumstances in the immediate vicinity, reliant upon expositional angels to descend into the frame and tell her about the greater world outside. The Paranormal Activity movies have no metaphysical ambitions beyond the interior of a suburban home, and at the end of the day, for all its apocalyptic, Manhattan-obliterating scale, Cloverfield really is just a story about a bunch of trust fund douchebags having a bad night.