From within the data bridge settings area you'll have the ability to view logs from the last 7 days:

Each time that a data bridge sync runs and data is copied from Property Hive to the installed theme, a log will be created.

In the dropdown of logs you can see the date and time that the sync began, and the date and time that it completed.

If the end time is '0000-00-00' this could indicate that the sync is currently running, or that an error occurred and it failed (e.g. maybe it timed out).

Selecting a log will show a breakdown of what happened during the sync: