Due to the frequent unauthorized access by commercial drones to Critical Infrastructures (CIs) such as airports and oil refineries, the US-based Federal Avionics Administration (FAA) recently published a new specification, namely RemoteID. The aforementioned rule mandates that all Unmanned Aerial Vehicles (UAVs) have to broadcast information about their identity and location wirelessly to allow for immediate invasion attribution. However, the enforcement of such a rule poses severe concerns on UAV operators, especially in terms of location privacy and tracking threats, to name a few. Indeed, by simply eavesdropping on the wireless channel, an adversary could know the precise location of the UAV and track it, as well as obtaining sensitive information on path source and destination of the UAV. In this paper, we investigate the trade-off between location privacy and data utility that can be provided to UAVs when obfuscating the broadcasted location through differential privacy techniques. Leveraging the concept of Geo-Indistinguishability (Geo-Ind), already adopted in the context of Location-Based Services (LBS), we show that it is possible to enhance the privacy of the UAVs without preventing CI operators to timely detect unauthorized invasions. In particular, our experiments showed that when the location of an UAV is obfuscated with an average distance of 1.959 km, a carefully designed UAV detection system can detect 97.9% of invasions, with an average detection delay of 303.97 msec. The UAVs have to trade-off such enhanced location privacy with a non-negligible probability of false positives, i.e., being detected as invading while not really invading the no-fly zone. UAVs and CI operators can solve such ambiguous situations later on through the help of the FAA, being this latter the only one that can unveil the actual location of the UAV.