When I set up a trade with a person and we message each other, is there any way to get notifications to show up somewhere (or turn them on if I don't)

As of now there isn't a way to receive email notifications when receiving new messages - only for trade proposals, address confirmations, and cancellations.

Alright, I wasn't sure if there was a way to get messages sent as emails and whatnot.

Getting notifications about unread trade messages is on the list of things to do, thanks for bringing it up :-)