In this paper, we are concerned with the stochastic susceptible-infectious-susceptible (SIS) epidemic model on the complete graph with $n$ vertices. This model has two parameters, which are the infection rate and the recovery rate. By utilizing the theory of density-dependent Markov chains, we give consistent estimations of the above two parameters as $n$ grows to infinity according to the sample path of the model in a finite time interval. Furthermore, we establish the central limit theorem (CLT) and the moderate deviation principle (MDP) of our estimations. As an application of our CLT, reject regions of hypothesis testings of two parameters are given. As an application of our MDP, confidence intervals with lengths converging to $0$ while confidence levels converging to $1$ are given as $n$ grows to infinity.