We study odd entanglement entropy (odd entropy in short), a candidate of measure for mixed states holographically dual to the entanglement wedge cross section, in two-dimensional free scalar field theories. Our study is restricted to Gaussian states of scale-invariant theories as well as their finite temperature generalizations, for which we show that odd entropy is a well-defined measure for mixed states. Motivated from holographic results, the difference between odd and von Neumann entropy is also studied. In particular, we show that large amounts of quantum correlations ensure the odd entropy to be larger than von Neumann entropy, which is qualitatively consistent with the holographic CFT. In general cases, we also find that this difference is not even a monotonic function with respect to size of (and distance between) subsystems.