Noise characteristics of state-of-the art light sources are crucial parameters in understanding their limitations towards quantum applications. This work describes a method to study the electrical noise transfer of current driver sources to the intensity noise of mid-infrared emission by commercial quantum and interband cascade lasers (QCLs and ICLs, respectively). A current driver with sub-shot electrical noise in a specific frequency range (up to 10 dB below the shot noise level) was developed for this purpose. This enables testing the performance of mid-infrared lasers when driven via such a quiet pump source. By using this novel current driver, we identify the fundamental noise of a QCL and an ICL, that is the laser intensity noise resulting solely from the internal dynamics of the laser under test. The proposed methodology allows us to retrieve the noise transfer function from current to light, showing that the main limitations in observing the quantum properties of the emitted photons come from laser excess noise and poor matching between laser and detection system in terms of bandwidth and optical power. From the analysis of the measured parameters, we highlight current technological limitations and suggest which key features should be optimized in mid-infrared systems for matching the performance required by quantum applications.