This paper studies distributed binary test of statistical independence under communication (information bits) constraints. While testing independence is very relevant in various applications, distributed independence test is particularly useful for event detection in sensor networks where data correlation often occurs among observations of devices in the presence of a signal of interest. By focusing on the case of two devices because of their tractability, we begin by investigating conditions on TYPE I error probability restrictions under which the minimum TYPE II error admits an exponential behavior with the sample size. Then, we study the finite sample-size regime of this problem. We derive new upper and lower bounds for the gap between the minimum TYPE II error and its exponential approximation under different setups, including restrictions imposed on the vanishing TYPE I error probability. Our theoretical results shed light on the sample-size regimes at which approximations of the TYPE II error probability via error exponents became informative enough in the sense of predicting well the actual error probability. We finally discuss an application of our results where the gap is evaluated numerically, and we show that exponential approximations are not only tractable but also a valuable proxy for the TYPE II probability of error in the finite-length regime.