The Port Authority, along with other New York Authorities like the MTA and Thruway Authority, is an entity called a "public benefit corporation".
Public benefit corporations are created by the the governnment State of New York to run various services like public transportation, highways, bridges and hospitals. They are quasi-governmental organizations that generally conduct their business under the same rules that government entities are supposed to adere to.
There are hundreds of Authorities in New York State, and like everything else connected to NYS government, authorities are generally rife with patronage, inefficiency and in some cases, corruption. It is generally understood that the NYS tends to hide its true debt and the true extent of its deficit spending by borrowing money through authorities, because authorities can float their own bonds.
Authorities, like publicly-traded corporations are required to make their financials available to the public. The PANYNJ's financials can be found here:
The Port Authority of New York and New Jersey - Financial Information
However, as a citizen, there is not much you can do about things like toll hikes because authorities are technically are run as independent entities