The development of reliable abstractions, models, and characterizations of biochemical communication channels that propagate information from/to biological cells is one of the first challenges for the engineering of systems able to pervasively interface, control, and communicate through these channels, i.e., the Internet of Bio-N ano Things. Signal transduction pathways in eukaryotic cells are important examples of these channels, especially since their performance is directly linked to organisms' health, such as in cancer. In this paper, a novel computational approach is proposed to characterize the communication performance of signal transduction pathways based on chemical stochastic simulation tools, and the estimation of information-theoretic parameters from sample distributions. Differently from previous literature, this approach does not have constraints on the size of the data, accounts for the information contained in the dynamic pathway evolution, and estimates not only the end-to-end information propagation, but also the information through each component of the pathway. Numerical examples are provided as a case study focused on the popular JAK-STAT pathway, linked to immunodeficiency and cancer.