High-density alkali vapors in the spin-exchange-relaxation free (SERF) regime are used in the highest-sensitivity optically-pumped magnetometers, atomic gyroscopes, and searches for physics beyond the standard model, and have been used to produce record numbers of entangled particles. Their power derives from a counter-intuitive effect (the SERF effect) in which spin-exchange collisions -a decoherence mechanism at low density -act to preserve coherence at high density.Here we present a first-principles analysis of the noise spectra of alkali vapors in and out of the SERF regime, and predict additional non-intuitive features, with potential to further improve the sensitivity of SERF media, and which must be taken into account in their use in quantum optical applications. Studying the process of spin-noise spectroscopy (SNS), we derive analytic formulae for the observable noise spectra, and for the correlation functions among different hyperfine components, which give additional insight into the spin dynamics. The analytic results indicate a variety of distortions of the spin-noise spectrum relative to simpler models, including a broad spectral background that mimics optical shot noise, interference of noise contributions from the two ground state hyperfine levels, noise reduction at the spin-precession frequency, and "hiding" of spin-noise power that can introduce a systematic error in noise-based calibrations, e.g. for spin-squeezing experiments.