Suppose I have a two-qubit circuit and have to measure the expectation value of an operator formed by the sum of Kroeneker products of Pauli Matrices (X, Y, Z, and I), such as
$$\hat O= aX\otimes X + bX\otimes Y + cY\otimes Z + d Z\otimes I.$$
To me seems clear that measuring $X\otimes X$ is just to put Hadarmad gates on both qubits and measure them. What about the "crossed" products such as $X\otimes Y$ or $Z\otimes I$?
It also seems that we can measure simultaneously products that commute such as $X\otimes I$ and $I \otimes X$. How the gates used to perform the measurements on one or another product can be handled to properly perform the simultaneous measurement?