Currently also happening to me. I don’t know when it started due to me never really using the Chat System. I recently started using the Roblox chat again to message someone. Note this isn’t due to adding someone new. It just randomly started happening. I’m thinking it might be a connectivity issue to Roblox servers and their chat system via chrome. Or just a problem with chrome entirely. Don’t quote me on this though. Privacy settings aren’t an issue and I’m also able to be chatted to.
Update
I uninstalled Google Chrome and reinstalled it. I went straight into Roblox and logged in, and the chat is working. I didn’t log-in or sync.
Update 2
I logged in and sync’d, not having any problems yet. I’ll update with a response if it stops working in the next couple of minuets.
Update 3
I just woke up and it has stopped working once again. It looks like reinstalling chrome without syncing first was just a hot-fix for a second.