The two types of life will compete with each other and among themselves for resources.
Even though the two life forms have different biochemistries, they presumably need similar raw materials: reduced carbon for example, to build their bodies, water, reduced nitrogen, sunlight, etc. Just as on Earth creatures with similar biochemistries compete with each other, on your world creatures with different biochemistries will also compete with each other. Suppose I am an earth fungus in the soil, soaking up nutrients. Maybe some of the nonearth things are my neighbors in the soil, doing the same thing. Those jerks. If one of my kind evolves some toxin to poison them (as fungi use pencillin to poison their competitors the soil bacteria) there will be more of me and less of them.
Suppose I am a cow. My rumen turns out to be a nice place for nonearth microbes to live. Now if I eat some nonearth plants, my commensal nonearth microbes will break them down inside my body. Some of the resulting molecules are usable by my cow self, and my commensals get the rest. Or vice versa, with earthlife microbes acting as commensals for nonearth creatures.
The received knowledge for now is that archaebacteria are a different form of microbial life that was outcompeted by eubacteria around the time of the Great Oxygenation. There are still plenty of archaebacteria but they are now relegated places where oxygen is low or the environment is otherwise super harsh - salty, hot, etc. Maybe that will happen to one or the other type of life in your world.
Life will be life. Life forms will reproduce, evolve and compete.