TBH, I don't go on the chat thing so I don't know how to help you. About a week or so ago (I think) JA said something about a new chat module or something coming due to a contributor. Maybe they're in the midst of implementing it. Maybe JA can shed some light on the subject.
The current chat is dumb and the new chat should be installed this week. (Courtesy of a generous Mugger who donated funds.)
Your alerts are not false, but it can seem that way on mobile devices. Once you click ok, you can click on the little chat bubble in the main menu to be taken to the chat page and it should be obvious who is messaging you. I believe you'll see a red bubble with a message count and their name. You can also change your settings there and make yourself invisible so people stop messaging you.