Large language model (LLM) personalization aims to tailor model behavior to individual users based on their historical interactions. However, its effectiveness is often hindered by two key challenges: the \textit{cold-start problem}, where users with limited history provide insufficient context for accurate personalization, and the \textit{biasing problem}, where users with abundant but skewed history cause the model to overfit to narrow preferences. We identify both issues as symptoms of a common underlying limitation, i.e., the inability to model collective knowledge across users. To address this, we propose a local-global memory framework (LoGo) that combines the personalized local memory with a collective global memory that captures shared interests across the population. To reconcile discrepancies between these two memory sources, we introduce a mediator module designed to resolve conflicts between local and global signals. Extensive experiments on multiple benchmarks demonstrate that LoGo consistently improves personalization quality by both warming up cold-start users and mitigating biased predictions. These results highlight the importance of incorporating collective knowledge to enhance LLM personalization.