Speech Overlap Detection
Most human vowels produces a series of harmonic spectral peaks that resolve to a single F0 pitch (plus some broad spectrum noise).  If you can't fit a single F0 pitch to the series of the spectral peaks, then you might try testing the hypothesis that the spectrum resolves to 2 or more f0 pitches, which would imply more than one speaker talking with an overlap of vowel sounds in their words.

Detecting an overlap between a vowel and someone else's consonant, without lots of false positives, seems like it might be far more difficult.