Sure, if there is one source of beeps, then no beeps would be conspicuous, but when there are many things all beeping, having one of them stop beeping may not be so obvious at all.
A simple technological solution seems appropriate here. Have each machine that needs to work check in every couple seconds with each of two centralized monitoring machines. Have each monitoring machine alert the nurse desk if a monitored machine stops checking in. This is silent and has no single point of failure.