Yes, with the SYNCING.NET you need to be connected to the internet for synchronization! Because of SYNCING.NET uses a peer-to-peer service for synchronization, that means a direct connection between at least two computers, it have to be at least one more computer online to get changes from other participants. (This is not needed if you checked the option SYNCING.NET “Configuration -> Options -> General Settings” “Enable Store and Forward”.)
The computer has also to be online if all participants are connected in the same local network (LAN). The peer-to-peer service is always trying to create the fastest connection between the computers, if this is possible over the local network, the data transfer will not go over the internet.