What privileges do off-duty pilots have in their own airline?
This is airline specific, so there's no single true answer: but I'll give an overview of what's "normal practice" for many airlines:

Airline staff will typically be bumped to First/Business class when there are available seats, although they're not usually top of the list.

In a typical airline, frequent flyers will be bumped first (particularly those with a lot of business class miles but who happen to be travelling economy right now): it's a way of rewarding loyal/frequent customers.

Once the frequent flyers are all moved up, though, the seats will often be filled with staff or contractors. Typically deadheading/on duty staff first (ie those who are flying home from a previous flight that they were working on), followed by off duty staff and their families.

This is airline specific, however - some airlines will bump everyone on the frequent flyer program  until Business is full, leaving no space for staff. Others will not move any passengers up (to discourage business class flyers deliberately trying to "game" the system by booking economy on quiet flights knowing that they'll likely be moved up), leaving more space for staff. Some airlines will not bump staff at all, or will only do so for deadheading staff (those who are flying because of work, not for their own holiday etc).

There's a lot of variation on things like whether your family will be bumped with you, or whether only higher-level contractors will be moved up. I used to work for a company which worked with airlines, and typically our directors would be moved up to Business while the lackeys were left in Economy