Of course it is only a theory, but a very plausible one given the frequency of rock transfer between planets and the toughness of microbes that allow them to survive in space for long time periods (both empirical facts).
There is really no way of proving transpermia - even if we to travel to Enceladus and found microbial life there how would we ever prove how it got there.
If we find DNA based microbial life it could certainly be amazing evidence in favor of transpermia. We would at least compare to DNA on earth and analyze similarities. Even plant and animal DNA are very similar (double helix shape)
Actually the easiest way to find life is to just look for DNA (or even better RNA) in the environment of interest. This is basically the way life is found in some of the amazing environments here on earth like the "lost city [0] or far under the earth [1].
Because of the technical simplicity and sensitivity of looking for DNA, any probe sent to any planet or moon would look for DNA first, basically assuming transpermia is true.
> given the frequency of rock transfer between planets
What is this frequency, exactly? My impression was that all planets are constantly bombarded with rocks from space (not from other planets), but that it would be extremely surprising for a rock to e.g. leave Earth and end up on Mercury.
There is really no way of proving transpermia - even if we to travel to Enceladus and found microbial life there how would we ever prove how it got there.