Privacy policies and legislation all play a role in keeping your rights of privacy in a good place.
The law in question is the Gramm-Leach-Blily Act, or GLB, as it is often called. This was enacted by Congress in July of 2001. It gives power to financial institutions to conduct business in many important ways.
Your personal information would include any information that can be identified as your particular information. It could be your name, address, social security number, internet banking transactions you have made in your name, and other information. It could also be any information that can be guessed by having some of this information.
If you opt-out, you are saying to the company that you do not want your information to be shared with other companies that are not affiliated with the bank. Under GLB, internet banking services still have the option of sharing your information with affiliates and marketing partners, but they should inform you that they are doing so.
Another problem is that some internet banking companies have made it very difficult to opt-out of having their information shared with unrelated businesses. It should be the easiest with online banking. Indeed some internet banking companies have online forms to fill out so that you can be finished with it in short order.
However, some internet banking sites make it nearly impossible to opt-out. They make you go to an 800 number, or write for a mail-in form, or they may not have opt-out at all. These are in the minority, but you should watch for them when comparing online banks.