We investigate the problem of placing monitors to localize node failures in a communication network from binary states (normal/failed) of end-to-end paths, under the assumption that a path is in normal state if and only if it contains no failed nodes. To uniquely localize failed nodes, the measurement paths must show different symptoms (path states) under different failure events. Our goal is to deploy the minimum set of monitors to satisfy this condition for a given probing mechanism. We consider three families of probing mechanisms, according to whether measurement paths are (i) arbitrarily controllable, (ii) controllable but cycle-free, or (iii) uncontrollable (i.e., determined by the default routing protocol). We first establish theoretical conditions that characterize network-wide failure identifiability through a per-node identifiability measure that can be efficiently evaluated for the above three probing mechanisms. Leveraging these results, we develop a generic monitor pl...