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We highlight the challenges faced by non-native speakers when using AI writing assistants to paraphrase text. Through an interview study with 15 non-native English speakers (NNESs) with varying levels of English proficiency, we observe that they face difficulties in assessing paraphrased texts generated by AI writing assistants, largely due to the lack of explanations accompanying the suggested paraphrases. Furthermore, we examine their strategies to assess AI-generated texts in the absence of such explanations. Drawing on the needs of NNESs identified in our interview, we propose four potential user interfaces to enhance the writing experience of NNESs using AI writing assistants. The proposed designs focus on incorporating explanations to better support NNESs in understanding and evaluating the AI-generated paraphrasing suggestions.