Does a search engine own the searches people perform?

There is no promise made for not selling terms. As a matter of fact, that is how Google blew up to be so big. They sell key words. The way it works is when you type in a key word such as "cars" companies such as BMW, FORD, MERCEDES pay to be at the top of the list when searches come back. All the items on the right of a google search result page are the paid ads. So yes, people make a profit off of words.