Overwhelmingly, the answers I've seen to this question on Reddit and elsewhere are anecdotal, so I would love to read any answers supported by strong research. However, I recognize that designing studies to answer this question are probably challenging due to sample size concerns, confounding, selection biases, etc.
A few important qualifiers to this question:
(1) I am specifically referring to primary and secondary education, not post-secondary education.
(2) I recognize that "life outcomes" is vague, but my goal was to keep the scope broad. Things that come to mind when I think of "life outcomes" which could be impacted by school type include, but are not limited to: substance use disorders/mental illness in childhood or adulthood; non-psychiatric illness in adulthood; expected lifetime wealth; expected lifetime career satisfaction; expected marital/relationship satisfaction; etc.
(3) I'd be open to comparisons between children who attend "average" private schools vs. those who attend "average" public schools... OR other comparisons, such as children who attend "average" private schools vs. those who attend "above-average" public schools. Again, I recognize that what constitutes an "average" school, or an "above-average" school, is vague, but I'd be open to any number of different operationalizations of these constructs (e.g., student-teacher ratios, AP classes offered, number of extracurriculars offered, etc.).