All Roku devices (and Roku TVs) should be able to route audio through the app to attached or wireless headphones as well. This might work with Share Audio on iPhones so you can have two people listening at once, but I've never tried.
last time I tried it , I found Roku audio via the app to be very laggy and sound cutting out often. This is despite having good wireless signal and no lag from anything else that uses wifi for steaming.
that's too bad. it worked well for me, but I never used it regularly.
I think some higher-end Rokus can route audio through their remote control, to plugged-in headphones.