Star Wars Battlefront – Can Trophies Be Earned in a Private Match?

There is no way to do it (at least for now). You are required to be online to view trophies. Quote:

The PlayStation 4 will not allow users to look at their Trophies or Friends lists when the console is offline, Polygon has learned using a retail unit received for review purposes.

The PS4 still doesn't sync Trophies in the background — every time you load up your list of virtual accomplishments, the console takes a moment to sync with Sony's servers. Perhaps because of that, you can't view your Trophies list if you're not connected to the internet. On PS3, you can scroll through your Trophies whenever you want, and you can sync them the next time you're online. You can definitely earn Trophies while playing PS4 games offline, but if you try to look at your list when you've disabled the console's internet connection, you'll get the following error message: "[Connect to the Internet] is turned off. (NW-31484-0)"

The trophies cannot be viewed offline with the PS4 like with the PS3 (you cannot toggle between online and offline), but you can still earn trophies by being offline. You will just need to connect online to sync them to your PSN account.