This paper focuses on the problem of distributed composite hypothesis testing in a noisy network of sparsely interconnected agents in which a pair of agents exchange information over an additive noise channel. The network objective is to test a simple null hypothesis against a composite alternative concerning the state of the field, modeled as a vector of (continuous) unknown parameters determining the parametric family of probability measures induced on the agents' observation spaces under the hypotheses. A recursive generalized likelihood ratio test (GLRT) type algorithm in a distributed setup of the consensus+innovations form is proposed, in which the agents update their parameter estimates and decision statistics by simultaneously processing the latest sensed information (innovations) and information obtained from neighboring agents (consensus). This paper characterizes the conditions and the testing algorithm design parameters which ensure that the probabilities of decision errors decay to zero asymptotically in the large sample limit.