Hi @nicolatoledo.dev !
Receive realtime data in a web application require websocket (web protocol for setting up a bidirectionnel canal between a client and a server so the server can inform the client when a new message is posted in the chat for example)
We do not support websocket on weweb for now, you will need to implement your own plugin with websocket capability.
I see Xano has a plugin to enable an external chat service : Xano - Ably Realtime Chat Extension - YouTube
But it looks like its only to do a public chat, and I see no mention of websocket 
Another solution to exploit this Xano plugin would be to perform a request to the xano database every x seconds to refresh the messages list. But I think we miss an action to setup this type of workflow but you can probably do it with javascript.
Do not hesitate if you have any other questions!