We characterize the Stein-exponent of a distributed hypothesis testing scenario where two sensors transmit information through a memoryless multiple access channel (MAC) subject to a sublinear input cost constraint with respect to the number of channel uses and where the decision center has access to an additional local observation. Our main theorem provides conditions on the channel and cost functions for which the Stein-exponent of this distributed setup is no larger than the Stein-exponent of the local test at the decision center. Under these conditions, communication from the sensors to the decision center is thus useless in terms of Stein-exponent. The conditions are satisfied for additive noise MACs with generalized Gaussian noise under a p-th moment constraint (including the Gaussian channel with second-moment constraint) and for the class of fully-connected (where all inputs can induce all outputs) discrete memoryless multiple-access channels (DMMACs) under arbitrary cost constraints. We further show that for DMMACs that are not fully-connected, the Stein-exponent is larger and coincides with that of a setup with zero-rate noiseless communication links from either both sensors or only one sensor, as studied in [1].