Both checks are performed via the Network tab of your browser's DevTools. In Google Chrome, you can access the DevTools by:
Pressing the F12 key;
Pressing the Command+Option+C (Mac) or Control+Shift+C (Windows, Linux, Chrome OS) key combination;
Clicking the Customize and control Google Chrome menu on the top right of the browser and then selecting More Tools > Developer Tools.
Once the DevTools sidebar is open, click on the Network tab. More information about the Google Chrome DevTools can be found in Google's Chrome DevTools documentation.
Validate the Google Ads Segment Synchronization
Navigate to the Network tab of the DevTools and refresh the page. You will see a lot of requests, search for usersync.datatrics.com.
If things are set up correctly, you will see a line for each segment and predictive field synchronization. Enter one of those terms in the search bar (like believer, as shown in the image above) and you should see requests directed to Google URLs (and possibly others).
More information can be found in the Using Datatrics audiences in Google Ads article.
Validate the Facebook Ads Segment Synchronization
Navigate to the Network tab of the DevTools and refresh the page. You will see a lot of requests; search for datatrics facebook.
If things are set up correctly, you will see requests going to Facebook URLs.
More information can be found in the Using Datatrics audiences in Facebook Ads article.