I will have someone open Photoshop on another computer; will they be able to see my files?

If the file is saved to disk, it goes nowhere. No one from the outside can access your computer (that would probably fall under the heading of spyware and be illegal).

Generally, people make far too much out of this "cloud" thing. It's just an online storage service, like any other online storage service out there. You don't need to use it if you don't want to. If you don't, Photoshop works just like it always has - only the licensing and payment method has changed..

Can you provide us more info please? How can someone open your Photoshop on another computer? Are you talking about Tem Viewer or similar app? 

When saving files you have option to choose where to save, in cloud or on local machine.

Photoshop is responsible for what it does when you open file and save, especially in PSD file format. Photoshop can not do things on your computer/OS nor it can hide something while using it.