When we first arrived on Alonissos 11 years ago there wasn't any WiFi available on the island. Within a couple of years funds were made available from the EU for an initiative called Teleaccess which put WiFi into various locations on the port area of Patitiri. On the harbour front in Patitiri there was a sign in the window of the Port Police office which is shown below. Most of the area on the harbour front had WiFi access via the Teleaccess system. You could then have a drink in one of the harbour front bars and access the Internet from your laptop. The Teleaccess centre was just passed the National Bank of Greece on the opposite side of the road.
Now on Alonissos virtually every café, bar, taverna, apartments or hotel has free WiFi available for you to access whether you are staying there or just visiting. If you are not resident somewhere which has WiFi access then buying a drink or having a snack is still the courteous way of thanking the establishment for the use of their connection to the Internet.