It'll be a Poisson distribution.
If the spread is too wide then the chance of spatiotemporally cohabiting civilizations of distinct origins will be diminishingly small.
But I don't actually think that the planetary parameters that allow for replication are (in any way) specific to earth.
That’s presumably true, but it means there could have been a very large number of losers not just 1. Aka if extremely primitive life showed up once on earth then it might be a rare event, if it happened 100 trillion times then most planets with the right conditions will have some form of life.