Yes, the module definitely needs to be on the aircraft
§ 89.115 Alternative remote identification. A person operating an unmanned aircraft that is not a standard remote identification unmanned aircraft may comply with the remote identification requirement of § 89.105 by meeting all of the requirements of either paragraph (a) or paragraph (b) of this section. (a) Remote identification broadcast modules. Unless otherwise authorized by the Administrator, a person may operate an unmanned aircraft that is not a standard remote identification unmanned aircraft if all of the following conditions are met:
(1) Equipage. (i) The unmanned aircraft used in the operation must be equipped with a remote identification broadcast module that meets the requirements of § 89.320 and the serial number of the remote identification broadcast module must be listed on an FAA-accepted declaration of compliance.
And the other requirements are:
§ 89.320 Minimum performance requirements for remote identification broadcast modules. Remote identification broadcast modules must meet the following minimum performance requirements:
(a) Take-off location. The remote identification broadcast module must be capable of determining the take-off location of the unmanned aircraft.
(b) Time mark. The time mark message element must be synchronized with all other remote identification message elements.
© Self-Testing and monitoring. (1) Prior to take-off, the remote identification broadcast module must automatically test the remote identification functionality and notify the person manipulating the flight controls of the unmanned aircraft system of the result of the test. (2) The remote identification broadcast module must continuously monitor the remote identification functionality from takeoff to shutdown and must provide notification of malfunction or failure to the person manipulating the flight controls of the unmanned aircraft system.
(d) Tamper resistance. The remote identification broadcast module must be designed and produced in a way that reduces the ability of a person to tamper with the remote identification functionality.
(e) Error correction. The remote identification broadcast module must incorporate error correction in the broadcast of the message elements in § 89.315.
(f) Interference considerations. The remote identification broadcast module must not interfere with other systems or equipment installed on compatible unmanned aircraft, and other systems or equipment installed on compatible unmanned aircraft must not interfere with the remote identification equipment.
(g) Message broadcast. (1) The remote identification broadcast module must be capable of broadcasting the message elements in § 89.315 using a non-proprietary broadcast specification and using radio frequency spectrum compatible with personal wireless devices in accordance with part 15 of title 47, Code of Federal Regulations, where operations may occur without an FCC individual license.