Training models on a carefully chosen portion of data rather than the full dataset is now a standard preprocess for modern ML. From vision coreset selection to large-scale filtering in language models, it enables scalability with minimal utility loss. A common intuition is that training on fewer samples should also reduce privacy risks. In this paper, we challenge this assumption. We show that subset training is not privacy free: the very choices of which data are included or excluded can introduce new privacy surface and leak more sensitive information. Such information can be captured by adversaries either through side-channel metadata from the subset selection process or via the outputs of the target model. To systematically study this phenomenon, we propose CoLA (Choice Leakage Attack), a unified framework for analyzing privacy leakage in subset selection. In CoLA, depending on the adversary's knowledge of the side-channel information, we define two practical attack scenarios: Subset-aware Side-channel Attacks and Black-box Attacks. Under both scenarios, we investigate two privacy surfaces unique to subset training: (1) Training-membership MIA (TM-MIA), which concerns only the privacy of training data membership, and (2) Selection-participation MIA (SP-MIA), which concerns the privacy of all samples that participated in the subset selection process. Notably, SP-MIA enlarges the notion of membership from model training to the entire data-model supply chain. Experiments on vision and language models show that existing threat models underestimate subset-training privacy risks: the expanded privacy surface leaks both training and selection membership, extending risks from individual models to the broader ML ecosystem.