A common way to handle such 1:1 chat rooms is to generate the room URL based on the user ids. As you already mention, a problem with this is that either user can initiate the chat and in both cases they should end up in the same room.
You can solve this by ordering the user ids lexicographically in the compound key. For example with user names, instead of ids:
var user1 = "Frank"; // UID of user 1
var user2 = "Eusthace"; // UID of user 2
var roomName = 'chat_'+(user1<user2 ? user1+'_'+user2 : user2+'_'+user1);
console.log(user1+', '+user2+' => '+ roomName);
user1 = "Eusthace";
user2 = "Frank";
var roomName = 'chat_'+(user1<user2 ? user1+'_'+user2 : user2+'_'+user1);
console.log(user1+', '+user2+' => '+ roomName);
<script src="https://getfirebug.com/firebug-lite-debug.js"></script>
A common follow-up questions seems to be how to show a list of chat rooms for the current user. The above code does not address that. As is common in NoSQL databases, you need to augment your data model to allow this use-case. If you want to show a list of chat rooms for the current user, you should model your data to allow that. The easiest way to do this is to add a list of chat rooms for each user to the data model:
"userChatrooms" : {
"Frank" : {
"Eusthace_Frank": true
},
"Eusthace" : {
"Eusthace_Frank": true
}
}
If you're worried about the length of the keys, you can consider using a hash codes of the combined UIDs instead of the full UIDs.