I have setup Firebase authentication in my game for Android. I need to implement a feature in my Android game, where a user can challenge another player. I need to implement it this way: The user is shown the list of all users registered on Firebase on the Game. To any user, he can tap on a button for respective user to send a challenge. When, he taps, a push notification is to be recieved by that user. When he taps on the notification, he is redirected to a specific part of the game. Maybe a specifc scene for now and he is shown there the Challenger's name. I have explored the Firebase documentation, but it does not provide information for client-client push notifications. Maybe there are other options available like GameSparks and OneSignal. But I don't know if there is a requirement of own Server for implementing it.
Don't send the notification directly. That isn't possible. However, you can deal with the notifications like this: One user sends the challenge to the other. I'd assume it could be as simple as a string saying that it is a challenge and a couple of strings with information about the sender. Then, all you have to do is tell the receiving end to notify the user every time it receives a challenge. This way, all the code is local, and the information being sent is nice and simple.
In short, send the other device the challenge, which then prompts the user with a notification. The rest is all handled locally. Hope this helps!