Large language models (LLMs) have demonstrated the ability to generate user-specific text with high stylistic fidelity. However, the personal data that enables such personalization frequently embeds demographic, cultural, and stylistic markers that raises concerns about stylometric re- identification. This paper investigates whether reducing identifiable stylistic signals affects personalization in text generation by LLMs. We introduce a controlled framework to isolate stylometric signals in LLM personalization using the LaMP-7 Twitter benchmark. Experiments on 250 sampled users compare two settings: paraphrasing conditioned on the original profile and paraphrasing conditioned on an anonymized converted profile in which demographic identifiers, cultural references, personal details, and informal linguistic cues have been systematically neutralized. Outputs are assessed by two independent LLM judges and a complementary human evaluation. Our pairwise evaluation shows that outputs conditioned on original profiles are nearly indistinguishable from human-authored ground truth, indicating that modern LLMs can closely reproduce an author's writing style with sufficient fidelity. In contrast, preference for model outputs with anonymized profiles drops to 13.0% on average, while semantic context preservation remains high at 94.8%. A study with human evaluators confirms the same pattern. These findings reveal a clear privacy-personalization trade-off and highlight the need for privacy-aware personalization methods that retain meaning while suppressing identifying stylistic signals.