What it means
Local Network permission lets an app discover or interact with devices on the same local network. Apple requires a prompt because nearby-device information can reveal details about your environment. Denying it does not necessarily block the app's internet connection, though a feature such as finding a casting target may fail. System services like AirPlay can have their own behavior, so the result must be tested in the app concerned.
A real-world example
A weather app asks for Local Network access at first launch, with no visible nearby-device function. The user declines and still receives forecasts. A speaker-control app asks during setup and cannot find the speaker until the user enables that specific permission.
What to do
Read the request in context. Deny first if the purpose is unclear, test the feature, then allow only the relevant app under Settings, Privacy & Security, Local Network if needed. Revoke old grants after devices or apps change.
The distinction that matters
Local Network permission is separate from location, Bluetooth, camera and router firewall rules. Allowing it is not proof the app is trustworthy, and denying it does not prevent every form of data collection. It is a focused control over one path to nearby devices. A VPN setting or cellular connection does not automatically explain the permission prompt; check the exact feature the app was trying to perform.
Apple local-network privacy