In terms of "auditing the books", Bitcoins presents an interesting situation where in there everybody could be performing an audit of some group's finances simultaneously. While this sort of spoils the anonymity aspect of Bitcoins (and talked about elsewhere), shareholders could insist that every transaction involve the same address for an audit trail. If there are transactions which happen "off the books", it would be tantamount to embezzlement.
I was thinking about the whole business activity, not just as far as money is concern. Sales, acquisition of material, expenses, salaries, etc... All this can not be reduced in terms of monetary transactions.
If you read a typical corporate annual report, it usually is reduced to just monetary transactions eventually, usually in the form of assets vs. liabilities. Materials acquired, sales, capital expenditures, salaries, and more are usually laid out systematically as things for which the bottom line depend, and almost always there is a bottom line that everybody relates to, particularly with publicly trade companies. Either the company is making a profit, or it is not. An audit is usually performed by some very well trusted 3rd party auditing company (generally not a government agency) who certifies the results of the audit, but the "net worth" of most public companies can usually be seen as a published number. I think for a group using Bitcoins in a similar fashion such "bottom line" figures will also be important to their investors.
I personally have problems with those companies who have in their charter the explicit clause that "the purpose of this company is to maximize profits and increase shareholder equity" as there are many other possible reasons to organize a company besides pure profit. Still, if you grabbed the corporate charters of almost every company on the NYSE I'm sure you would find either that statement explicitly or a variation of it in some form. CEOs in particular are interested in returning a profit for their shareholders as it is a dominant part of their job and the explicit reason why they got hired in the first place.
The trick when investing is to take into consideration perhaps some of those non-monetary aspects that you are talking about, but that is something which is also a part of the "goodwill" and "trust" that the company over time brings to its investors, employees, customers, and the community at-large. It tends to be something that markets in general are very good at evaluating in general, particularly a large auction market that has reduced barriers to entry. It also takes freedom as in free exchange of information that is critical. A group which is hiding their activities from their investors is going to earn bad will and eventually be rejected from the market, contaminating other enterprises they get involved with.