We study how the amount of correlation between observations collected by distinct sensors/learners affects data collection and collaboration strategies by analyzing Fisher information and the Cramer-Rao bound. In particular, we consider a simple setting wherein two sensors sample from a bivariate Gaussian distribution, which already motivates the adoption of various strategies, depending on the correlation between the two variables and resource constraints. We identify two particular scenarios: (1) where the knowledge of the correlation between samples cannot be leveraged for collaborative estimation purposes and (2) where the optimal data collection strategy involves investing scarce resources to collaboratively sample and transfer information that is not of immediate interest and whose statistics are already known, with the sole goal of increasing the confidence on an estimate of the parameter of interest. We discuss two applications, IoT DDoS attack detection and distributed estimation in wireless sensor networks, that may benefit from our results.