This contribution analyzes healthcare co-design processes through participant observation of seven Health Budget co-design tables at ASL Roma 6 (October-December 2024). Thematic analysis of collected materials identifies four critical dimensions: organizational differentials among Third Sector entities based on collaborative capacity rather than size; tensions between administrative-accounting and generative logics requiring political mediation skills; impact evaluation as a collective process of building consensus on value criteria; implicit collaboration costs often unrecognized in budgets. Results show how co-design constitutes a negotiation space among actors carrying different institutional logics, requiring specific investments in collaborative capacity to ensure model sustainability. The study contributes to the debate on enabling conditions and concrete criticalities accompanying shared administration implementation in local contexts, providing insights for policy makers, service operators, and researchers interested in public administration innovation processes.