The rise of Artificial Intelligence as a Service (AIaaS) democratizes access to pre-trained models via Application Programming Interfaces (APIs), but also raises a fundamental question: how can local models be effectively trained using black-box models that do not expose their weights, training data, or logits, a constraint in which current domain adaptation paradigms are impractical ? To address this challenge, we introduce the Black-Box Distillation (B2D) setting, which enables local model adaptation under realistic constraints: (1) the API model is open-vocabulary and trained on large-scale general-purpose data, and (2) access is limited to one-hot predictions only. We identify that open-vocabulary models exhibit significant sensitivity to input resolution, with different object classes being segmented optimally at different scales, a limitation termed the "curse of resolution". Our method, ATtention-Guided sCaler (ATGC), addresses this challenge by leveraging DINOv2 attention maps to dynamically select optimal scales for black-box model inference. ATGC scores the attention maps with entropy to identify informative scales for pseudo-labelling, enabling effective distillation. Experiments demonstrate substantial improvements under black-box supervision across multiple datasets while requiring only one-hot API predictions. Our code is available at https://github.com/yasserben/ATGC.