"Private" not private
According to Ars Technica, Verizon saves your deets and sends them to Bing (but they promise to delete them some day, honest):
1. Your IP address, search query, and user agent are transferred over HTTPS to Verizon servers. The user agent generally includes data about the browser, operating system, and type of device and app you're using to make the search.
2. Verizon derives your city-level location data from your IP address and then sends your IP address, user agent, search query, and location data to Microsoft's Bing "so that the actual search request can be made through their search engine."
3. Bing provides the search results to Verizon, and then Verizon's automated process "work[s] with our Search Partners to provide you with contextual advertisements and/or search results." Verizon describes the "search partners" vaguely as "certain companies providing search result optimization input" and says they "are not provided with your personal data."
4. Verizon will store your IP address for four days "for the purpose of network traffic protection" and then permanently delete the IP address.
5. Bing will continue to store the IP address, search query, and user agent, also for network traffic protection. After four days, Bing "obfuscates the IP address."