NancyAK,
The federal folks require that annual independent audits be performed on organizations that receive over a certain dollar amount. Many of the the state grants, if over a certain amount also require that an audit be performed annually. Under the IRS laws/regulations, a non-profit has to file a 990 once a year. Any person can walk in off the street and ask the organization to see or have a copy of their 990. 990's are like a income tax report in away but has more info on it. More and more organizations are posting their 990's online for public inspection as well. In addition to that, non-profits who are required to have annual audits are required to allow the public to view them with the exception of the "Letter to Management".  In addition to that, there is an obscure law in Alaska under civil code regarding the right of members of non-profits to be able to go to the business office and view financial and other records (not personnel or client). 
Boards have fiduciary duties.