When you self-host Flarum, you have full control over the software and data.
The Flarum Foundation doesn't collect any information, and doesn't have any backdoor into self-hosted forums. There is no telemetry baked into the software at all. We only have some limited statistics about the number of ZIP and Composer downloads. Composer reports download statistics to Packagist to update the number of time each extensions and dependencies are installed or updated, which you can probably opt out from. You can also run your own Packagist mirror or use a proxy if you want to completely hide your server IP from Packagist.
Third-party extensions might collect additional information, you would need to check with each vendor. Unfortunately there is currently no way to automatically know if an extension collects or shares data, we rely on developers self-reporting. I don't know of any open-source extension that integrates telemetry. A number of extensions interact with third-party APIs, in which case it's pretty obvious you are bound to the privacy policy of those cloud services. But they also don't usually have any way to read your private information, unless the integration is intentionally bi-directional.
The Flarum emojis are loaded from a CDN (not controlled by Flarum), but you could disable the Emoji extension to just use native UTF8 emojis if you don't want this feature. There are also a few other formatting features like code highlighting that relies on scripts hosted on CDNs.
On the forum itself, you can use the Flarum permission system to configure which information is readable by guests, registered members or privileged groups.
If you have questions about specific features or extensions, feel free to ask. Please include your php flarum info output or a link to the extension (or post directly in that extension's discussion on this forum) so we know which extension(s) you are referring to.