This paper studies recursive composite hypothesis testing in a network of sparsely connected agents. 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. Specifically, under the alternative hypothesis, each agent sequentially observes an independent and identically distributed time-series consisting of a (nonlinear) function of the true but unknown parameter corrupted by Gaussian noise, whereas, under the null, they obtain noise only. Two distributed recursive generalized likelihood ratio test type algorithms of the \emph{consensus+innovations} form are proposed, namely $\mathcal{CIGLRT-L}$ and $\mathcal{CIGLRT-NL}$, in which the agents estimate the underlying parameter and in parallel also update their test decision statistics by simultaneously processing the latest local sensed information and information obtained from neighboring agents. For $\mathcal{CIGLRT-NL}$, for a broad class of nonlinear observation models and under a global observability condition, algorithm parameters which ensure asymptotically decaying probabilities of errors~(probability of miss and probability of false detection) are characterized. For $\mathcal{CIGLRT-L}$, a linear observation model is considered and upper bounds on large deviations decay exponent for the error probabilities are obtained.