If you use google for searching the web, the text you have entered into the search bar is stored on your computer, and your computer only. It's stored in the temporary internet folder, the same as cookies, etc.
However, I've heard some rumours about Spyware being attatched somehow to search bars, in browsers, and the ability for the site to monitor the things you search for. You can delete the list that shows in the drop down menu by deleting all temporary internet files and history, in your browser. Tools > Options.