I've tried playing through the Mac GOG Galaxy and downloading the separate offline installation. Neither will launch. Is it because Catalina is so new? Any help is appreciated!

So, the problem is that MacOS Catalina dropped support for 32-bit applications. Privateer requires DOSBox to run, and for MacOS they've used Boxer—a MacOS DOSBox wrapper. Unfortunately, Boxer hasn't (officially) been updated to 64-bit.

DOSBox was updated with 64-bit MacOS support in version 0.74-2, and since it can be run on MacOS you could download DOSBox and try to get the game running yourself. If you want to try that route, Here's a link to the official DOSBox download page. Version 0.74-3 is the latest at the time of this writing.

Alternately, users have gotten Boxer built with 64-bit support. So you could try using a rebuilt Boxer for an easier drop-in replacement. You can find the GitHub thread for 64-bit support here. It hasn't made it into an official build yet, but there hasn't been an "official" build in years, so don't hold your breath waiting for one.

Side Comment: GOG should have updated all of the Boxer-based game pages to inform MacOS users of this problem... and still should.

Excellent. Thank you very much!

Sure thing! Write back with details if you get it working :)

Looks like GOG has updated it to work with the current iteration of MacOS Catalina.