Critical observability is a property of cyber-physical systems to detect whether the current state belongs to a set of critical states. In safety-critical applications, critical states model operations that may be unsafe or of a particular interest. De Santis et & x00A0;al. introduced critical observability for linear switching systems, and Pola et & x00A0;al. adapted it for discrete-event systems, focusing on algorithmic complexity. We study the computational complexity of deciding critical observability for systems modeled as (networks of) finite-state automata and Petri nets. We show that deciding critical observability is: 1) NL-complete for finite automata, i.e., it is efficiently verifiable on parallel computers; 2) PSPACE-complete for networks of finite automata, i.e., it is very unlikely solvable in polynomial time; and 3) undecidable for labeled Petri nets, but becoming decidable if the set of critical states (markings) is finite or cofinite, in which case the problem is as hard as the nonreachability problem for Petri nets.