I know of various third-party IM clients which allow you to do this. Been a long-time user of Trillian myself, for which this was one of the cooler features back in the day. Allowed you to be notified of people typing, regardless of having an open conversation, and set up whether you wanted to send typing notifications always, only after having sent a initial message, or never. Was pretty cool to be able to say "Hi!" to someone just when they were about to message you. Most of my friends never figured how I did that, since their official clients never supported it so they didn't even know it could be a thing.

Haven't got Facebook myself, but I know Trillian fully supports Facebook chat and I suppose many other (free/open source) IM clients do as well. Such multi-network clients are in my experience great for slowly moving people away from pretty much any network, since they pretty much remove the distinction between them. I just add my friends and let the software figure out how exactly it gets the message to them. :)