Two MPPT controllers will fight each other trying to find max power point when connected to same PV array. If PV array power is greater than can be handled by one inverter then split into two arrays with each feeding separate inverter. If using separate batteries then there will no sharing of received PV power between AC split phases.
As to having separate batteries, if the inverters have cross control, when one shuts down, for any reason, the other will also shut down. This is a requirement if running 120vac inverters in series for 240vac split phase. Also, the cross control ensures they stay 180 degs phase relationship sync'd when running off grid or when grid goes down. You would not want your 240vac to drop half its supply when only one inverter shuts down or go willy-nilly phase drifting when grid goes down as that would corrupt 240vac output.
With separate batteries, whichever phase side is weakest link for capacity will set when system shuts down for depleted battery. If one side 120vac phase load is greater on weaker separate battery side than system capacity will suffer more.