Currently in osu! the only way to know someone sent you a PM through the website is to log on and not everyone does that when they play, I was thinking when you login to the client if you have an unread PM it will come up with text telling you have one and either have a little (!) on the screen or BancoBot sends you a PM telling you to check your inbox and give you a link to the pm, or the inbox if it has multiple.