Unless you post a message in a chat room other members of the room should not be able to ping your inbox. Otherwise you end up in a situation where you can come back to multiple inbox notices for a room you were just lurking in.
Chat room lurking is a way of checking out if one is interested in a room or not. A lot of people do it and never speak, and they may hang out in the room for a while intentionally (or unintentionally, if they don't realize they never actually signed out of it in the chat interface).
For people in the room, there is a balance between greeting people so they do not feel they are interrupting a conversation...and just ignoring them so they feel like outsiders who shouldn't interrupt. Outreach to a lurker can involve browsing their SO profile, and perhaps even multiple people will have something to say to the new "potential chatter".
However, there's not a good reason for the StackOverflow inbox to reflect the existence of these messages. They are ephemeral to the period of potential interest. A beep or notification to the lurker--only while they are present in the chat room--should suffice.
This has nothing to do with "shutting out lurkers". There are two categories of notifications in chat:
- beeps in the room while you're in it and
- those red flags that show up in your "SO inbox" at the top of the site when you're not logged into chat that tell you about your mentions in question comments etc.
I'm merely suggesting eliminating the second category notifications for people who have never in a chat room's history spoken in it but just joined and left (or thought they left).