At startup it will check this URL: "
https://www.screenpresso.com/binaries/version4.xml" to check for updates. You can disable this via settings or group policy (GPO) but only with Screenpresso PRO since free versions requires the user to always use latest version of Screenpresso.
When the user uses a sharing feature (e.g. "Publish to Google Drive"), a HTTPS connection is used. Publish feature can be disabled via group policy.