The business doesn't pay its owners a cent. It pays its employees (who happen to be the owners) a salary.
In a business where profit is used to pay shareholders/owners you don't count the regular salaries as 'profit paid out' either. This is no different; any profit we have is reinvested (while financially stable, there has been no occasion where the profits exceeded the amount prudent to reinvest in growth or other factors).
What happens if the company makes more money than it pays salaries for some reason? I assume in this case and your description, the salaries are adjusted accordingly.
If so, then the word salary here is just another word for owner dividends, at least to some extent.
Those are referred to as "retained earnings" (occasionally: accumulated earnings, retained capital, or earned surplus), and accrue to the not-for-profit organisation's balance sheet.
Note that this may occur on a short-term basis given normal market variability, or over the long term The latter case is more interesting.
In the short term, such business profits simply accrue to accounts, and are typically held as cash or cash-equivalents.
Over the medium term, funds may be moved to various forms of generally-liquid assets (government bonds, equities, etc.), as part of the organisation's money-management strategy.
In the long term ... the organisation gets to decide how to invest those resources. There are several options:
- Increase wages, as you suggest. This is of course only one option.
- Expand business. This might entail expanding or improving a location, opening additional location, going into multiple lines of business, or aquiring other organisations.
- Expand endowment. This is particularly the case for many educational institutions, and those endowments may range into the billions of dollars.
Generally you'll find these options discussed in a long term capital plan or similar strategy document.
For specific forms of business organisation, such as a co-operative business (which may or may not be a not-for-profit organisation), there may also be dividend payouts to members. My understanding is that these are not equivalent to stock dividends, though I'm hazy on details.
What you don't see for non-profits, however, is a mandated and frequently automatic form of return to shareholders, through either stock dividends or inflating the value of a joint stock issue. That cuts two ways: not-for-profits don't have the financial drag of spilling out money to investors, but they also typically have less access to financial capital markets.
In a business where profit is used to pay shareholders/owners you don't count the regular salaries as 'profit paid out' either. This is no different; any profit we have is reinvested (while financially stable, there has been no occasion where the profits exceeded the amount prudent to reinvest in growth or other factors).