The Anatella software in-itself does not access the network.
However, some ODBC drivers (for example: the MySQL ODBC drivers) and some OleDB drivers are accessing their database through a network connection. For these reason, you should configure your firewall to allow Anatella to access your network (or at least, to allow access to the required database).
There are also some specific Anatella Actions that are accessing the network when used (This is a normal an expected behavior): Such Actions are:
Query SMS Delivery Status, Geocoding & Reverse Geocoding, TCP-IP Receive table, Get Email, InfoOnIp, Send Email, TCP-IP Send table, Send SMS HTTP, Send SMS SMPP, Curl Download, Send a Message to Slack, etc.