Currently, you can only see your finishing time for the entire game and of course, you will see your time if you beat your Personal Best on an individual map. I personally think it would be useful to see your finishing time for each map, no matter if you beat your Personal Best or not. The fact that there already is a Personal Best feature proves that a time tracking feature is already in the game, the time would just have to displayed. If people are worried about these messages clogging up the chat, it might be best to send them only to the player who finishes a map so that everyone only sees their own times.
Of course in connection with this, it should definitely be possible to see your Personal Best without actually having to beat it. Meaning that there should be a lobby item or at least a command so that you can view your Personal Bests. Again, since they have to keep track of them anyway, why not add a feature to show them to the player? I know this has been suggested plenty of times before but I wanted to bring it up since it is in very close connection to my suggestion.