The magnetic dissipative droplet is a strongly nonlinear wave structure that can be stabilized in a thin film ferromagnet exhibiting perpendicular magnetic anisotropy by use of spin transfer torque. These structures have been observed experimentally at room temperature, showcasing their robustness against noise. Here, we quantify the effects of thermal noise by deriving stochastic equations of motion for a droplet based on soliton perturbation theory. First, it is found that deterministic droplets are linearly unstable at large bias currents, subject to a drift instability. When the droplet is linearly stable, our framework allows us to analytically compute the droplet's generation linewidth and center variance. Additionally, we study the influence of nonlocal and Oersted fields with micromagnetic simulations, providing insight into their effect on the generation linewidth. These results motivate detailed experiments on the current and temperature-dependent linewidth as well as drift instability statistics of droplets, which are important figures-of-merit in the prospect of droplet-based applications.