  Joint Gaussian measurements of two quantum systems can be used for quantum
communication between remote parties, as in teleportation or entanglement
swapping protocols. Many types of physical error sources throughout a protocol
can be modeled by independent Gaussian error channels acting prior to
measurement. In this work we study joint Gaussian measurements on two modes
$\mathsf{A}$ and $\mathsf{B}$ that take place after independent single-mode
Gaussian error channels, for example loss with parameters $l_\mathsf{A}$ and
$l_\mathsf{B}$ followed by added noise with parameters $n_\mathsf{A}$ and
$n_\mathsf{B}$. We show that, for any Gaussian measurement, if $l_\mathsf{A} +
l_\mathsf{B} + n_\mathsf{A} + n_\mathsf{B} \geq 1$ then the effective total
measurement is separable and unsuitable for teleportation or entanglement
swapping of arbitrary input states. If this inequality is not satisfied then
there exists a Gaussian measurement that remains inseparable. We extend the
results and determine the set of pairs of single-mode Gaussian error channels
that render all Gaussian measurements separable.
