I think it's a fun question and I can kind of give a layman's answer. (though reading the above answer, there is some overlap).
It seems likely that "tiny", life or viruses could survive/hibernate for a long time if blown into space on space rocks, so it follows that life could travel to earth that way, but I think we could apply a modified version of the drake equation.
The Earth gets hit by comets or meteors every day but most of those burn up in the atmosphere. There is a temperature beyond which even bacteria and viruses have a hard time surviving. So, many of those meteors or bits of comet, I suspect, would lose any life they might have. Only a percentage could be the right size to both slow down enough while breaking apart to not impact at too high a speed and successfully deliver life to the surface or into the ocean. Like the real drake equation, we get to plug in our own numbers, but I suspect most impacts wouldn't successfully deliver life to the planet they impact. How many would? 1 in 10? 1 in 100? I'm not sure.
Now, what percentage of comets or meteors have life on them? This is a tough one, I'm pretty sure life can only form on a planet because you need (I would think), liquid water, atmosphere to maintain surface water and reduce evaporation and some kind of heat to drive the mixing of chemicals, perhaps lightning or undersea volcanic vents. Life likely cannot be formed in space or in a nebula. It can be blown into space off a planet with life, but it can't form in space. That likely reduces the number of meteors or objects in space that even have the possibility of carrying life. I hate to guess this one, cause who knows, but 1 in 1000? 1 in 10,000? Most of the meteors, asteroids and comets in space have probably formed without the possibility of carrying life.
There are some simple organic molecules in space, alcohol for example, but there's an enormous chasm between organic molecules and even the simplest life.
Some planets are very difficult to blow debris off of. Earth with an atmosphere and an escape velocity of some 25,000 miles per hour needs a very big impact to lose material into space. Mars, with a very thin atmosphere and lower escape velocity is much easier. Of some 60,000 recognized meteorites, 124 have been identified from Mars. (Source) and Mars is the easiest one. Venus, with it's thick atmosphere and Mercury, being as close to the sun it is, adding the suns escape velocity to it's own, are much harder. Material being blown off the gas giants is even less likely, so Mars is the easiest plant to potentially lose material where it could land on Earth or other planets. Even if we use the Mars Meteor ratio of 400 to 1, which is, I think, too high, it's still a very low percentage of objects in space that potentially came off a planet or moon and carry life with them.
A 3rd point worth bringing up is repetition. Lets say, just for fun, that Europa and Enceladus both have life in their under the surface oceans and both planets regularly eject jets of life filled water into space, that freezes and carries basic life into orbits around Jupiter / Saturn and some of those life carrying bits of ice make their way on occasion to other planets. If this happens it probably happens more than once and any bits of life that may come to earth from Europa or Enceladus, if it happens at all, probably happens more than once. The first introduction of invasive species, even simple ones, can be problematic, but after the first, the planet should have adapted. That's one of the mathematical quirks. One Andromeda strain can in theory be a huge problem, but repeated Andromeda strains from the same source, the risk drops with each impact. What you need, for a dangerous situation to happen is a new introduction of unfamiliar that's never happened before. That's a pretty specific requirement. Mars may have once had life, and if it had, life on Mars probably landed on earth many times over 4 plus billion years.
a 4th point, space is full of radiation. Even viruses will eventually die exposed to the radiation of space. A space rock, carrying life from another solar-system light-years from earth has a very long journey and a very small target. Bits of frozen ice or rock probably travel from other solar systems into ours quite rarely over very long periods of time, and, as I pointed out above, most of those probably won't have life.
This is just a kind of "shoot from the hip" sort of answer, but I think you would need a perfect storm of events for your scenario to actually happen. statistically, I suspect it's a very rare occurrence and not something we need to worry about on a human time frame.