You will receive notification of chat replies in your inbox after a sufficiently long period (15-20 minutes), provided you don't acknowledge them by clearing the notification in the chat window or sending new messages.
An overview of what factors come into play when determining if you receive an inbox notification for a chat message can be found in this post:
To use you as an example:
- @TonyMeyer, i.e. an exact match will cause a notification if you've
been in the room in the past seven days.
- @Tony, i.e. a first name
match (to be precise, a word boundary match) will cause a notification
if you've been in the room in the past two days.
- @Ton will notify you
if you've been in the room in the past two days and there's no other
user for whom this would be a better match, e.g. someone called "Ton"
who has been there within the past seven days (that's 1.) or someone
called "Ton Ynnuf" who has been there within the past two days (that's
- An explicit reply to one of your messages (like this one) will always
cause a notification.
In addition, these notifications will only be delivered to your global
inbox if you haven't seen them within a reasonable amount of time (~20
minutes), where "seen" is defined as one of the following:
- visiting a direct permalink to the message in question,
- acknowledging it through clicking the little number that's popping up on you profile
picture in the chat room when you have replies,
- or saying something in
the room in question.