Here is an outline of the private bidding process. Your first time out you should give yourself plenty of time. You will need bidding documents and general conditions in addition to your normal construction documents.
* develop a list of suitable contractors, any one of which you would be happy to work with, minimum three, maximum of six
* prepare and send them an invitation to bid
* make all the documents available through your repro shop's online plan room at their own cost
* set a bid due date and time that gives the contractors three weeks from when drawings are available
* have the bids due at your office or the client's; make it clear that the bid opening is private
* issue a formal addendum if the need for more info or clarifications arise; this can be done by email
* open the bids shortly after the deadline (with or without your client as they wish), tabulate the results, verify completeness, determine the best bid if there are alternates to consider, and report your recommendation to the client
* make sure there is a contract
The bidding documents you will need are the Invitation to Bid, Instructions to Bidders, Bid Form, General Conditions and Sample Contract or identification of what is intended. The client's attorney or insurance agent should provide some of the information. There is an AIA document that helps collect the needed information - G-603, I believe.
Update: There are two related articles.
Preparing To Bid A Project