School-age children on the autism spectrum who are minimally verbal (MVAS) use a limited repertoire of spontaneous communicative spoken words and reveal large heterogeneity in cognitive functioning. Despite the challenges to form peer social engagement posed by their unique social-communicative profile, few interventions have targeted peer interactions in the MVAS population. This study explored predictors of individual differences in treatment response among 54 school-age minimally verbal autistic children (8-16 years) following an RCT "school-based peer social intervention" (S-PSI) that compared two peer-oriented intervention modalities (conversation versus collaboration) versus a waitlisted control group. We examined autistic-symptom severity, age, verbal and nonverbal IQ, executive functions, and sensory-processing profile for their contribution to children's ability to form relevant spontaneous communication exchanges with a peer partner. Main findings revealed that larger deficits in sensory-processing (sensory-avoidance and sensory low-registration) and in executive functions contributed to greater growth in "relevant" (i.e., adequately attuned, participatory, reciprocal) communication following both interventions, but not for the waitlisted controls. Additionally, older participants with lower verbal and nonverbal IQ improved communication's relevancy more after the conversation intervention, whereas the collaboration intervention was more beneficial for younger participants. Lower autistic-symptom severity contributed to larger growth in relevancy for all groups. By identifying individual-level predictors of spontaneous, attuned, participatory, and reciprocal (i.e., "relevant") communication exchanges with peers, we optimized S-PSI personalization for this uniquely nonverbal, heterogeneous MVAS population. These new channels for tailoring peer interventions to better meet individuals' needs may result in reduced social isolation and loneliness and enhanced well-being.