They wouldn't have ownership in both clubs until the contract is signed. There is nothing in the regulations to stop anyone who is involved with a club bidding on another, they would have to resign their position at their original club if the sale went through and they then had ownership. It is the only scenario I can see that would require absolute anonymity be maintained but is feasible.
I admit it is just a supposition but does answer the questions being asked as to why someone would pull out. It would not be because of the negative comments or the fact that a £2 million bid was being made, it would be the fact that they feared the reprisals if their identity was revealed and the bid failed.
As I say it is just me speculating as to the reasons the bidder pulled out and whether it is right or not I don't know but it is just as valid as most of the speculative rumours being given for other things that have happened with the club.
Edit:
Sorry forgot to answer the second part of your response regarding how they would know who the other bidders are. I am sure they would have a good idea of who would be in the running, just as Nixon will, they will have been researching and using other contacts to obtain an insight into who they are up against. That's normal business practice when acquiring any assets.
The problem anyone needing anonymity would have is that if any other bidder decides to leak information there will always be a fear of what they know and what they are prepared to leak.