This paper identifies the Stein-exponent of distributed detection when the sensor communicates to the decision center over a discrete memoryless channel (DMC) subject to one of three stringent communication constraints: 1) The number of channel uses of the DMC grows sublinearly in the number of source observations n; 2) The number of channel uses is n but a block-input cost constraint is imposed almost surely, which grows sublinearly in n; 3) The block-input constraint is imposed only on expectation. We identify a dichotomy in the Stein-exponent of all these setups depending on the structure of the DMC's transition law. Under any of these three constraints, when the DMC is partially-connected (i.e., some outputs cannot be induced by certain inputs) the Stein-exponent matches the exponent identified by Han and Kobayashi and by Shalaby and Papamarcou for the scenario where communication is of zero-rate but over a noiseless link. We prove our result by adapting Han's zero-rate coding strategy to partially-connected DMCs. In contrast, for fully-connected DMCs, in our scenarios 1) and 2) the Stein-exponent collapses to that of a local test at the decision center, rendering the remote sensor and communication useless. %To prove this result, we propose new converse proofs relying on change of measure arguments. In scenario 3), the sensor remains beneficial even for fully-connected DMCs, however also collapses compared to the case of a partially-connected DMC. Moreover, the Stein-exponent is larger when the expectation constraint is imposed only under the null hypothesis compared to when it is imposed under both hypotheses. To prove these results, we propose both new coding strategies and new converse proofs.