Speech Source Separation (SSS) or Audio Source Separation (ASS) can be seen as a specialized version of source separation. I mention these expressions under which one can find additional works. One acceptation of the "Cocktail Party Problem" is the task of hearing/recovering one specific sound of interest in a complex environment (one-source versus all), while your objective seems more ambitious: unmixing all sources (another common term). The 2015 paper The cocktail-party problem revisited: early processing and selection of multi-talker speech reviews speech-related and perception issues.
The possibility of their identification depends a lot on the quality and quantity of observations, and the formation model of the observed signals. If the model is linear and instantaneous, this is already complicated. Single channel source separation is a specific topic of interest. In non-linear environments, when the number of sources is greater than the observations, when convolutive effects or echoes happen, when noise is difficult to tackle, blind source separation techniques are unlikely to succeed without additional priors and ancillary information/models.
Humans use their binaural features a lot in this context. Therefore, using spatial information from similar sensors is useful, but this can be insufficient. Indeed, there is a whole domain of Audio-Visual Speech Source Separation because:
The separation of speech signals measured at multiple microphones in
noisy and reverberant environments using only the audio modality has
limitations because there is generally insufficient information to
discriminate fully the different sound sources. Humans mitigate this
problem by exploiting the visual modality which is insensitive to
background noise and can provide contextual information about the
Combining audio and visual sensors is an instance of increasing the diversity in the sources: the more the sources can avoid to overlap in the recorded domain, the highest the chance of separation. PCA is very limited to that respect because it is too-strongly-tied to correlation and orthogonality. It is linear, nonparametric and cannot (easily) incorporate prior knowledge. It can estimate decorrelated components, up to a rotation. In other words, suppose that we two are talking. A PCA could detect the following two sources: 1) your voice minus mine 2) your voice added to mine, not what you'd expect.
However, as a whitening or compression method, PCA can be used as a preprocessing to other methods like Independent Component Analysis (ICA), see for instance A. Hyvarinen, J. arhunen, E. Oja, Independent Component Analysis, 2001.
Additional speech features, like sparsity in a time-frequency domain (see Sparse Representations for the Cocktail Party Problem, 2006), more informed (less-blind) source separation (see From blind to guided audio source separation: How models and side information can improve the separation of sound) can help.
Finally, the link Deep Learning Machine Solves the Cocktail Party Problem may provide some pointers to the use of machine learning and artificial intelligence.