Wow, that's... actually pretty sensical. The science part of me rejects it for having no evidence. But the pragmatic part of me would put its money underneath the theory. The reason
The bigger question is: how did the first Protozoa form? The usual primordial soup theory of random chance?
One good thing about this theory is that if there are 10^7 habitable planets at any given moment, each of them are "trying to produce life." So the chance that life gets to Earth increases multiple fold.
One problem for this theory is that if it takes a thousand years for Earth to get protozoa from another planet, then we would expect to have gotten multiple different types of Protozoa. But there is basically no variety on Earth: it's all carbon, and 99% uses oxygen (a few use sulfur).