Truck platooning, an autonomous driving technology, reduces fuel consumption and emissions by organizing heavy-duty vehicles (HDVs) into convoys. While single-carrier platooning is feasible, cross-carrier implementations present challenges due to privacy concerns between competing carriers and third parties. This paper presents a real-time, privacy-preserving coordination framework for cross-carrier platooning. The framework safeguards sensitive itinerary data against both peer carriers and third-party service providers. Secure multi-party computation techniques are employed to ensure that planning data remains private, while collaborative decision-making enables effective coordination without the need for a centralized third party. A distributed model predictive control approach dynamically updates truck plans at hubs to optimize platooning opportunities. The framework is evaluated through large-scale simulations using real-world-inspired data, demonstrating its practicality. Results indicate a minor reduction in cost-saving performance but no significant computational overhead from privacy-preserving mechanisms compared to predictive coordination with the third party, highlighting an effective balance between privacy and coordination effectiveness.