The Trustee Sale is a verbal auction. It can be held "on the courthouse steps" but it can also be held on a walkway outside of an office building, or on a narrow balcony in front of the trustee office's entry door.
The published and recorded Notice of Sale will state the time and location where the Trustee Sale is to be held. It is never held at the property-in-forclosure.
One of two things will happen at the Trustee Sale:
An individual will make an acceptable bid and the property will be sold to that person.
Or no one will make an acceptable bid, and ownership of the property will revert back to the lender.
If ownership reverts to the lender, the property then becomes part of that lender's inventory of "Real Estate Owned", which is where the term "REO" comes from.