This is a really simple one to answer: Only you.
That is, there is no-one who can look at your searches, save people who have access to your account. PatSnap doesn't record your searches to anywhere outside of your own username. If you give your details to someone else, or you share access, then they'll be able to look at your history. Unless of course, you delete it.
The Company Administrator can change a setting to turn off the recording of the search history company wide. This can be done by going to the Admin Settings and turning on Incognito Mode. This function is currently only available for the Analytics product.