listener_class.dart

class RecommendationStream {
  RecommendationStream() {
    ...
    _notificator.outEvent.listen((event) {
      print('ЭТОТ ТЕКСТ ПЕЧАТАЕТ ДВАЖДЫ');
      if (event == ProfileScreenEvents.UPDATE_LOCAL) {
        send();
      } else if (event == ProfileScreenEvents.UPDATE_REMOTE) {
        sendNew();
      }
    });
  }
  
  ...
}

screen_1.dart

...
    onTap: () {
      Navigator.pushNamed(
        context,
        SettingsScreen.routeName,
        arguments: args,
      ).then((value) {
        context
            .read<MainSourceData>()
            .notificator
            .inEvent
            .add(ProfileScreenEvents.UPDATE_REMOTE);
      });
    },
...

screen_2.dart

...
  Navigator.pushNamedAndRemoveUntil(
    context,
    UserProfileScreen.routeName,
    ModalRoute.withName(StartScreen.routeName),
    arguments: args,
  );
  Timer(
    Duration(seconds: 3),
    () {
      ProfileScreenNotification _notificator = ProfileScreenNotification();
      StreamSink<ProfileScreenEvents> profileUpdater = _notificator.inEvent;
      profileUpdater.add(ProfileScreenEvents.UPDATE_REMOTE);
    }
  );
...

singleton_broadcast_stream.dart

import 'dart:async';

enum ProfileScreenEvents { UPDATE_LOCAL, UPDATE_REMOTE }

class ProfileScreenNotification {
  static final ProfileScreenNotification _instance = ProfileScreenNotification._();

  ProfileScreenNotification._();

  factory ProfileScreenNotification() {
    return _instance;
  }

  StreamController<ProfileScreenEvents> _controller =
      StreamController<ProfileScreenEvents>.broadcast();

  StreamSink<ProfileScreenEvents> get inEvent => _controller.sink;

  Stream<ProfileScreenEvents> get outEvent => _controller.stream;

  void dispose() {
    _controller.close();
  }
}