 Solution

Prepare an ancilla qubit in $|0⟩$. To measure $X\otimes X$, apply a Hadamard gate to each data qubit, apply a controlled-NOT gate from each data qubit to the ancilla, measure the ancilla in the computational basis, and apply a Hadamard gate to each data qubit again. The ancilla records the parity of the two rotated computational-basis bits, so outcome $0$ corresponds to eigenvalue $+1$ and outcome $1$ to eigenvalue $-1$. The data register is projected by $[I+(-1{)}^{s}X\otimes X]/2$, so its complete post-measurement state is retained. For $Z\otimes X$, perform the same ancilla-assisted Pauli measurement but apply the basis-changing Hadamard gates only to the second data qubit.