I'm not sure that's how it works. As I understand it, the model predicts bubble nucleation, not state leakage, due to spatial inhomogeneities: a local energy fluctuation leads to a true-vacuum bubble forming and expanding.
However, let's leave aside the spatial inhomogeneity for the moment. As I understand it, tunneling in many worlds would result in a continuum of decayed worlds being continuously spawned, all with equal and infinitesimal probability, with the un-decayed one slowly decreasing in probability.
Assuming a Schrodinger cat-type experiment, with Eliezer being the cat, and assuming that Eliezer dies in every decayed world (not an unreasonable assumption if it's vacuum that decays), and assuming quantum immortality-type ontology (quite a number of assumptions), Eliezer will only ever perceive the surviving branch, with no measurable leakage.
The talk by Joseph Lykken (Lykken, ironically, is Norwegian for luck), making the science news rounds today, conjectures that Higgs will some day destroy the universe in a flash of a "true vacuum" expanding at light speed and destroying our "false vacuum", and everything else in it. Here is the original paper. The interesting part for me is not the idea, which is not at all new, but the related anthropic reasoning, which goes as follows:
That alternate universe would be "much more boring," Lykken said. Which led him to ask a philosophical question: "Why do we live in a universe that's just on the edge of stability?" He wondered whether a universe has to be near the danger zone to produce galaxies, stars, planets ... and life.
Cue Frost's Fire and Ice...