Estimating the quality of remote speech communication is a complex task influenced by the speaker, transmission channel, and listener. For example, the degradation of transmission quality can increase listeners' cognitive load, which can influence the overall perceived quality of the conversation. This paper presents a framework that isolates quality-dependent changes and controls most outside influencing factors like personal preference in a simulated conversational environment. The performed statistical analysis finds significant relationships between stimulus quality and the listener's valence and personality (agreeableness and openness) and, similarly, between the perceived task load during the listening task and the listener's personality and frustration intolerance. The machine learning model of the task load prediction improves the correlation coefficients from 0.48 to 0.76 when listeners' individuality is considered. The proposed evaluation framework and results pave the way for personalized audio quality assessment that includes speakers' and listeners' individuality beyond conventional channel modeling.