This answer can be improved. I do not have enough time now.
According to both Jewish Law and American Law (as far as I know) a sale is legally binding as long as there was a meeting of the minds (true agreement) and an exchange of consideration (payment) and an execution (kinyan-transaction took place).
Witnesses are not needed for a financial transaction to be valid and binding on the parties. Witnesses serve to enforce the truth of the contract/sale in case of a dispute. (An exception would be Jewish marriage where witnesses actually effect the marriage event. Without witnesses to a marriage, it didn't take place in Jewish Law). The sale of chametz is a simple financial issue and no witnesses are needed to accomplish the sale, Jewish or Gentile.
The use of witnesses at all, is an extra measure to give pomp, ceremony, and decorum to the transaction. It creates the air of official business. This is done because the Jewish Rabbis of Europe (from about the 1500's to 1900's) were divided about the truth of any such large scale transaction/sale of a community's chametz to Johnny Gentile. Many Rabbis said the whole thing is a scam. Johnny doesn't want the bread nor does he have the funds required to buy it all. The Jews don't know what they are selling and what they are being paid. The Gentile defaults every year like clockwork and receives a bottle of wine from the Rabbi. Its a joke! So the objection was that such sales are a mere charade of a formality. Many Jews today will not rely on selling chametz to a Gentile in order to buy it back after Pesach.
The Chasam Sofer was the main Rabbi who voiced the opinion that all such sales are definitely valid. Many Rabbis and communities follow this opinion because as long as the transaction was made properly, it works..period.
However, in deference to the objectors who would cast doubt on such sales, many things are done to show it is in fact a real business deal. Witnesses bolster this viewpoint.
In reality, they would only be needed if someone disputed the transaction after the fact. If that would happen, the two witnesses could surrender their share in the transaction. This sacrifice would allow them to testify in a Jewish court (Rambam hilchos Edus). An American court would still accept their testimony anyway but the judge would take their position in the deal into consideration when hearing their testimony and cross examination.
I hope this helps. Chag Kosher V'Sameach