Good question. The other person's Caller ID can show different messages. If you're using Yahoo! Phone Out, the number +1 (777) 777-7777 or "A Yahoo-owned number" is shown. Either way, your identity isn't revealed.
If you use the Yahoo! Phone In service, and have one Yahoo! Phone In number, the default setting will be that number, not "Anonymous." If you have more than one Yahoo! Phone In number, the default setting will be the last number purchased.
If you're using Yahoo! Phone In, you can choose "Anonymous" (or any of the numbers you have purchased). To set the Caller ID preference: