Panspermia, the idea that life started at some point off-Earth and only spread to Earth later, is a well-established idea, but I don't think I've ever seen anyone talk about the inverse, that life (in this solar system) started on Earth and has spread to basically everywhere else where it could survive between then and now.
The argument goes something like this: if there is a greater-than-infinitesimal probability of an organism somehow escaping Earth and remaining viable long enough to reach somewhere else in the solar system where it could thrive, it probably has happened already, and from there, the remaining potentially livable virgin habitats would be reached soon after, since these 'colonized' locations would also be generating their own panspermic events, possibly even more often than Earth due to a likely smaller gravity well.
Of course, this scenario is dependent on that 1st conditional. So, are there any reasonably probable scenarios for such to happen? Below are two possibilities:
Life could potentially hitch a ride in ejecta created by an asteroid impact. It is questionable whether anything could survive an impact sufficiently violent to create such ejecta, but, if it was possible and there was an ejecton that carried viable life, there is little question that it would reach the ejecton's destination (well, there is the question of time, it could be millions or even billions of years before it collides with anything, but there are indications that bacterial spores can survive for millions of years if sufficiently insulated from radiation, which a decently large ejecton would do). That's the other problem, however: any given ejecton destination is not likely to be able to support life. If the probability of an ejecton carrying viable life in the first place is also low, this doesn't seem like it would be a viable vehicle for reverse panspermia.
Life could be carried off Earth by solar radiation pressure. To be clear, life cannot remain viable when directly exposed to space, so this scenario is specifically talking about life inside of sufficiently small particles. The upside to this scenario is that it is happening all the time, increasing the probability of success massively. However, it is questionable that these particles could protect their spores for long enough reach their destination viable (do note that, due to the quantity of these particles, there will be some that reach livable destinations in centuries or less). There is also the fact that these particles would have trouble actually reaching their destination, since every potential habitat is covered by a nonlivable surface.
How would we test this hypothesis? Why, by going everywhere in the solar system where life could potentially thrive and looking for it, of course. If life is found in a lot of these places, and it is found to share an origin with Earth life (especially if it is found that it seems to diverge from Earth life after LUCA, which would solidly rule out the idea that they both had a shared origin off-Earth), then this is confirmed.
It would be fairly easy to tell if an example of non-Earth life had an independent origin. One possible tell would be if it was found in conditions Earth life couldn't possibly survive in, but there is only one place where such is even considered a possibility (Titan's surface). The main way would be a broad analysis of its features. Even if abiogenesis occurred in very similar conditions to how it did on Earth, it is highly unlikely that every innovation occurred in the exact same way; there would be extreme structural dissimilarities in what should be highly conserved elements, especially in large biomolecules. Hence, if such structural dissimilarities aren't observed, the non-Earth life would have to have a shared origin with Earth life.
As an aside, to address the more typical conception of panspermia: I don't think either are very likely. Wait, either? Well there are two versions of the idea. One being that life started somewhere, probably in interplanetary space, in the early solar system, and the other being that fully formed life from another solar system was seeded on Earth by chance.
My issue with the first one is that, well, it just seems unlikely. Both genetic and fossil evidence point to LUCA being around 4 billion years old. The features of early life make a lot of sense for an early Earth, such as the reverse citric acid cycle, affinity with water, getting energy from something that isn't sunlight. Earth has much more in the way of resources; early life probably would have exhausted whatever asteroid it had formed in fairly quickly. Early life would be even more vulnerable than current life to hitching a ride through space, since things like spore formation hadn't been innovated yet.
As for the second one: for one, it just moves the problem of the origin of life back. The issues highlighted above about hitching a ride through space would be immensely exacerbated by the much larger timescales needed for interstellar travel. Life has gotten more complex since LUCA, even when you only look at bacteria (hell, life gained three canonical amino acids—tryptophan, tyrosine, and methionine—well after LUCA), which wouldn't be expected from life with an already long evolutionary history.