Large Language Models (LLMs) are transforming geospatial artificial intelligence (GeoAI), offering new capabilities in data processing, spatial analysis, and decision support. This paper examines the open-source paradigm's pivotal role in this transformation. While proprietary LLMs offer accessibility, they often limit the customization, interoperability, and transparency vital for specialized geospatial tasks. Conversely, open-source alternatives significantly advance Geographic Information Science (GIScience) by fostering greater adaptability, reproducibility, and community-driven innovation. Open frameworks empower researchers to tailor solutions, integrate cutting-edge methodologies (e.g., reinforcement learning, advanced spatial indexing), and align with FAIR principles. However, the growing reliance on any LLM necessitates careful consideration of security vulnerabilities, ethical risks, and robust governance for AI-generated geospatial outputs. Ongoing debates on accessibility, regulation, and misuse underscore the critical need for responsible AI development strategies. This paper argues that GIScience advances best not through a single model type, but by cultivating a diverse, interoperable ecosystem combining open-source foundations for innovation, bespoke geospatial models, and interdisciplinary collaboration. By critically evaluating the opportunities and challenges of open-source LLMs within the broader GeoAI landscape, this work contributes to a nuanced discourse on leveraging AI to effectively advance spatial research, policy, and decision-making in an equitable, sustainable, and scientifically rigorous manner.