Chemically active systems such as living cells are maintained out of thermal equilibrium due to chemical events which generate heat and lead to active fluctuations. A key question is to understand on which time and length scales active fluctuations dominate thermal fluctuations. Here, we formulate a stochastic field theory with Poisson white noise to describe the heat fluctuations which are generated by stochastic chemical events and lead to active temperature fluctuations. We find that on large length- and timescales, active fluctuations always dominate thermal fluctuations. However, at intermediate length- and timescales, multiple crossovers exist which highlight the different characteristics of active and thermal fluctuations. Our work provides a framework to characterize fluctuations in active systems and reveals that local equilibrium holds at certain length- and timescales.