This is all speculative, but given the scenario, neither species would be able to completely wipe out the other, seeing how far away the origin of either one is from the other. Ultimately, both sapiens would need to develop alongside each other.
Early humans weren't very intelligent. I'm talking about Homo Sapiens here. Even with the capacity for higher thinking, they weren't too different from other animals, with only a very primitive language and grasp on tools, and they would be extremely territorial. Assuming human migration up to the point they meet is largely the same, the two species would have met in India between 80,000 and 120,000 years ago.
There are signs that Homo Sapiens and Homo Neanderthalensis (Neanderthals) weren't always on the best of terms, and probably cannibalized each other, like AndreiROM mentions, but the same is true for any differing tribe of Homo Sapiens, too. The appearance and similarities between two tribes didn't really factor in, and it was just a tribe mentality, similar to how modern day humans often still prefer their own race or country over another. Humans and Neanderthals also have left evidence that not all of their interactions have been violent. Depending on how your marsupial sapiens look and act, it's possible that the primate sapiens (humans) could either eventually develop some sort of mutual relationship, or all together try to eliminate each other—but it ultimately depends on the amount of resources. Considering the marsupials would come from Australia, it wouldn't be feasible for homo sapiens to outright eliminate them though.
Assuming that both sides survive first contact and there is a sizable population of marsupial sapiens on the Eurasian continent, the primate sapiens would go on to populate Europe and west Asia, while our marsupials could possibly hold onto East Asia, and even be the ones to migrate into the Americas, assuming there were similar pressures as in our own timeline. The separation probably won't be so clean, and there'd be tribes of both mixed in either region, which could either end up wiped out or tolerated long enough to develop.
If and when tribal societies evolve into more stable villages and cities, is probably when you can expect the relationship from both species to be much less extreme. There is the issue that the Silk Road might not actually exist, because even if both sides aren't constantly trying to eat each other, they still might not like each other enough to trade. This means that the benefits Europe had over Asia and the rest of the world (that eventually led to military domination), may not develop in the same way, and technological progress may stagnate for quite a while. The Black Death probably also won't happen in the same way that it did because diseases between the two species would rarely be compatible, so assuming Europe faces the same issues it was facing in our own timeline, they probably won't have the same dark miracle that reduced their population by so much and allowed the age of enlightenment to kick start.
Taking that into account, if Europe ever decided to cross the Atlantic, the Marsupials living in the Americas wouldn't suffer the same destruction by disease as the Native Americans did in our timeline.
What might happen with the marsupial sapiens that inhibit Australia and Asia? I can't tell. I don't know what sort of temperament they hold compared to humans, what sort of language they would develop, and so on. My assumptions on Europe are based on the idea that it would mostly be inhabited by humans, and there would be a pretty big buffer between them and the marsupials. However, it's easily likely that over time the Eurasian continent had both species mix around less cleanly than what my scenario implies. After all, it's not really likely that there'd be a hard wall between where one group or the other lives.
There are many different ways this scenario could play out, as you further advance their technology. There's also the issue of religion, and how the religion of one race would view the other, and what effect that would have on the future relations. There's no definite answer to this question, because we only have a sample size of 1 sapient animal.