Yes, the target device must have an internet capability enabled in order for TheTruthSpy to record the activities of your child or employee.
The software requires unrestricted GPRS internet on the phone and a data/internet plan, whether it be unlimited traffic or a specific amount of traffic.
The software may function over a wi-fi hotspot, given that the hotspot has no filters or firewalls in place that could interfere with the software.
But, no need to have an internet connection alwasy. Just some time to let the software flush data to the server where you can read it (at https://my.thetruthspy.com)