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In the realm of collaborative learning, extracting the beliefs shared within a group is a critical capability to navigate complex tasks. Inherent in this problem is the fact that in naturalistic collaborative discourse, the same propositional content may be expressed in radically different ways. This difficulty is exacerbated when speech overlaps and other communicative modalities are used, as would be the case in a co-situated collaborative task. In this paper, we conduct a comparative methodological analysis of extraction techniques for task-relevant propositions from natural speech dialogues in a challenging shared task setting where participants collaboratively determine the weights of five blocks using only a balance scale. We encode utterances and candidate propositions through language models and compare a cross-encoder method, adapted from coreference research, to a vector similarity baseline. Our cross-encoder approach outperforms both a cosine similarity baseline and zero-shot inference by both the GPT-4 and LLaMA 2 language models, and we establish a novel baseline on this challenging task on two collaborative task datasets---the Weights Task and DeliData---showing the generalizability of our approach. Furthermore, we explore the use of state of the art large language models for data augmentation to enhance performance, extend our examination to transcripts generated by Google's Automatic Speech Recognition system to assess the potential for automating the propositional extraction process in real-time, and introduce a framework for live propositional extraction from natural speech and multimodal signals. This study not only demonstrates the feasibility of detecting collaboration-relevant content in unstructured interactions but also lays the groundwork for employing AI to enhance collaborative problem-solving in classrooms, and other collaborative settings, such as the workforce. Our code may be found at: (https://github.com/csu-signal/PropositionExtraction).