This paper focuses on the problem of distributed composite hypothesis testing in a network of sparsely interconnected agents, in which only a small section of the field modeling parametric alternatives is observable at each agent. A recursive generalized likelihood ratio test (GLRT) type algorithm in a distributed setup of the consensus-plus-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. Finally, simulation studies are presented which illustrate the findings.