The Remote ID (RID) regulation recently introduced by several aviation authorities worldwide (including the US and EU) forces commercial drones to regularly (max. every second) broadcast plaintext messages on the wireless channel, providing information about the drone identifier and current location, among others. Although these regulations increase the accountability of drone operations and improve traffic management, they allow malicious users to track drones via the disclosed information, possibly leading to drone capture and severe privacy leaks. In this paper, we propose Obfuscated Location disclOsure for RID-enabled drones (OLO-RID), a solution modifying and extending the RID regulation while preserving drones' location privacy. Rather than disclosing the actual drone's location, drones equipped with OLO-RID disclose a differentially private obfuscated location in a mobile scenario. OLO-RID also extends RID messages with encrypted location information, accessible only by authorized entities and valuable to obtain the current drone's location in safety-critical use cases. We design, implement, and deploy OLO-RID on a Raspberry Pi 3 and release the code of our implementation as open-source. We also perform an extensive performance assessment of the runtime overhead of our solution in terms of processing, communication, memory, and energy consumption. We show that OLO-RID can generate RID messages on a constrained device in less than 0.16 s while also requiring a minimal energy toll on a relevant device (0.0236% of energy for a DJI Mini 2). We also evaluate the utility of the proposed approach in the context of three reference use cases involving the drones' location usage, demonstrating minimal performance degradation when trading off location privacy and utility for next-generation RID-compliant drone ecosystems.