The specifics of the title question are tricky, however.
It could also be likened to what happens when you put a powerful magnet next to a screen. That's a greatly simplified version of how I imagine the Shadow in the Warp works. The Tyranids emit their own signals, not dissimilar, but their sheer volume drowns out anything else, making our signal completely impossible to detect. Imagine every sentient being giving off a sort of radio signal, and those waves coalesce into what we call the Warp. If we assume that every Tyranid soul, barring psykers on both sides, is roughly equivalent to a human/Ork/Eldar soul in power, the psychic field (Tyranid-Warp) that causes the Shadow is astronomically more powerful than our own Warp. I explain the Tyranid ability to perfectly recycle their biomass by assuming that they use some kind of psychic energy - either relating to the organism's own "spark" or not - to handle the process, escaping entropy through use of magic (otherwise the energy comes from nowhere, and with Tyranids so intimately tied to magic as a race, it fits). We don't know what happens to this "spark" (for want of a better word) once the organism dies, but it would follow a pattern for Tyranids and for "our" Warp that the "spark" enters the Shadow-Warp again.
It would stand to reason that the Shadow in the Warp is essentially their own, alien, Warp (or rather, the Shadow is the noticeable effect on the Warp caused by the Tyranid-Warp). We can assume that they have some equivalent of a soul - a psychic spark of sorts - but unique. Truly alien, not humans-with-extras like Orks, Tau or Eldar. Like a lot of things relating to the Tyranids, the specifics of their supernaturalism are, well, alien.