  This article examines the problem of state estimation over multi-terminal
channels in an unreliable regime. More specifically, we consider two canonical
settings. In the first setting, measurements of a common stochastic source need
to be transmitted to two distinct remote monitors over a packet-erasure
broadcast channel. In the second setting, measurements of two distinct
stochastic sources need to be transmitted to a common remote monitor over a
packet-erasure multi-access channel. For these networked systems, we uncover
the fundamental performance limits in the sense of a causal tradeoff between
the estimation error and the communication cost by identifying optimal encoding
and decoding strategies. In the course of our analysis, we introduce two novel
semantic metrics that play essential roles in state estimation over broadcast
and multi-access channels. The first metric arising in the context of broadcast
channels is the dissemination value of information, which quantifies the
valuation of provisioning a piece of information to multiple receivers
simultaneously. The second metric arising in the context of multi-access
channels is the prioritization value of information, which quantifies the
valuation of provisioning a piece of information chosen from one out of
multiple transmitters. Our findings certify that the optimal encoding and
decoding strategies hinge on these semantic metrics.
