If you mean the programmer-visible on-chip state, you need the registers: A, X, Y, S, P, PC.
The contents of the stack are part of the contents of memory. If you want two processors to stay in sync they need to be executing the same program, seeing the same data, and the same interrupts at exactly the same times. The memory image they see would need to be the same, and that includes the stack.
As to how to set up A, X, Y, S, P and PC, you can only do it by executing code, so you'd need a way to execute a little more than half a dozen instructions.
If the two processors actually have different memories and peripherals, you need to get those into the same state too.
It would all make a lot more sense if you could say what you're thinking of doing, and a sketch of how you're thinking of doing it.
And some people will be wondering why you're doing it, so that might be worth a few words too!
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