Per my comment on the other answer, you should not use the Xcode capabilities UI for this. It will not correctly handle the two separate entitlement files in a Flutter project.
You need to add:
<key>com.apple.security.network.client</key>
<true/>
to macos/Runner/DebugProfile.entitlements and macos/Runner/Release.entitlements.
This is documented here and here.