This is one of many possibilities, but I consider it a long shot.

The most primitive life we have identified on Earth, the archeobacteria, include bacteria that cannot live in an oxygen atmosphere and that use methane to produce energy. That means there was an abundant source of methane in the early earth environment. There is no indication that methane was ever a constituent of any martian atmosphere. Given what we know about organosynthesis, it is far more likely to me that life could have appeared independently on two planets rather than sprung from a common source.

It is also possible that if there is any life on Mars, it could have gotten there from Earth by the same meteoritic impact and hitch hiking process. Some scientists still think it is possible that much of the organic processing that led to life on Earth happened off-Earth, and this could include even the partial assembly of DNA material, although I find this hard to believe. Planetary surfaces are far more stable and hospitable places for stitching together large molecules than the surfaces of dust grains, small asteroids and comets.

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