  Harnessing the advantages of shared entanglement for sending quantum messages
often requires the implementation of complex two-particle entangled
measurements. We investigate entanglement advantages in protocols that use only
the simplest two-particle measurements, namely product measurements. For
experiments in which only the dimension of the message is known, we show that
robust entanglement advantages are possible, but that they are fundamentally
limited by Einstein-Podolsky-Rosen steering. Subsequently, we propose a natural
extension of the standard scenario for these experiments and show that it
circumvents this limitation. This leads us to prove entanglement advantages
from every entangled two-qubit Werner state, evidence its generalisation to
high-dimensional systems and establish a connection to quantum teleportation.
Our results reveal the power of product measurements for generating quantum
correlations in entanglement-assisted communication and they pave the way for
practical semi-device-independent entanglement certification well-beyond the
constraints of Einstein-Podolsky-Rosen steering.
