In a short sale, the answer is it depends on the bank. There is no rule or law that says the bank has to require an arms length transaction, even in a short sale. Sometimes they ask for it and other times they don't. The banks in the last four short sales that I have closed did not require an "arms length" document to be signed. In theory, since they didn't require the document for closing a person who wasn't arms length could have bought the house. One way to get around the requirement at this point would be to propose a settlement which will circumvent most of the traditional short sale overhead - the process it tricky though so make sure you get someone to help you.
If you are talking about bidding at the foreclosure auction, commonly known as the sheriff's sale, the bidding is open to the public and anyone including the current owner can bid on the property. The down side is to win at the sheriff's sale you usually have to have the cash on hand - no mortgages here.
After the bank owns the house, having taken it back by foreclosure, they are free to sell it to whomever they please. I cannot concieve of a reason why the bank would require an arms length type transaction once they own it, and certainly have never seen that, but again it is up to each individual bank.
Whatever you do, make sure that you get a good agent involved in the process who has experience dealing with banks and distressed property situations.
Post back some additional comments or questions about your situation and I know that there are many of us here that would be more than happy to continue to guide you along the way.
Coldwell Banker Burnet - White Bear Lake/Forest Lake