We define two complementary approaches to monitor decentralized systems. The first relies on those with a centralized specification, i.e, when the specification is written for the behavior of the entire system. To do so, our approach introduces a data-structure that i) keeps track of the execution of an automaton, ii) has predictable parameters and size, and iii) guarantees strong eventual consistency. The second approach defines decentralized specifications wherein multiple specifications are provided for separate parts of the system. We study two properties of decentralized specifications pertaining to monitorability and compatibility between specification and architecture. We also present a general algorithm for monitoring decentralized specifications. We map three existing algorithms to our approaches and provide a framework for analyzing their behavior. Furthermore, we introduce THEMIS, a framework for designing such decentralized algorithms and simulating their behavior. We show the usage of THEMIS to compare multiple algorithms and verify the trends predicted by the analysis by studying two scenarios: a synthetic benchmark and a real example.