AI-based systems such as language models have been shown to replicate and even amplify social biases reflected in their training data. Among other questionable behaviors, this can lead to AI-generated text--and text suggestions--that contain normatively inappropriate stereotypical associations. Little is known, however, about how this behavior impacts the writing produced by people using these systems. We address this gap by measuring how much impact stereotypes or anti-stereotypes in English single-word LM predictive text suggestions have on the stories that people write using those tools in a co-writing scenario. We find that ($n=414$), LM suggestions that challenge stereotypes sometimes lead to a significantly increased rate of anti-stereotypical co-written stories. However, despite this increased rate of anti-stereotypical stories, pro-stereotypical narratives still dominated the co-written stories, demonstrating that technical debiasing is only a partially effective strategy to alleviate harms from human-AI collaboration.