Federated min-max learning has received increasing attention in recent years thanks to its wide range of applications in various learning paradigms. Similar to the conventional federated learning for empirical risk minimization problems, communication complexity also emerges as one of the most critical concerns that affects the future prospect of federated min-max learning. To lower the communication complexity of federated min-max learning, a natural approach is to utilize the idea of infrequent communications (through multiple local updates) same as in conventional federated learning. However, due to the more complicated inter-outer problem structure in federated min-max learning, theoretical understandings of communication complexity for federated min-max learning with infrequent communications remain very limited in the literature. This is particularly true for settings with non-i.i.d. datasets and partial client participation. To address this challenge, in this paper, we propose a new algorithmic framework called \ul{s}tochastic \ul{s}ampling \ul{a}veraging \ul{g}radient \ul{d}escent \ul{a}scent ($\mathsf{SAGDA}$), which i) assembles stochastic gradient estimators from randomly sampled clients as control variates and ii) leverages two learning rates on both server and client sides. We show that $\mathsf{SAGDA}$ achieves a linear speedup in terms of both the number of clients and local update steps, which yields an $\mathcal{O}(\epsilon^{-2})$ communication complexity that is orders of magnitude lower than the state of the art. Interestingly, by noting that the standard federated stochastic gradient descent ascent (FSGDA) is in fact a control-variate-free special version of $\mathsf{SAGDA}$, we immediately arrive at an $\mathcal{O}(\epsilon^{-2})$ communication complexity result for FSGDA. Therefore, through the lens of $\mathsf{SAGDA}$, we also advance the current understanding on communication complexity of the standard FSGDA method for federated min-max learning.