It also doesn't really matter if you colleagues or boss find out. What are they gonna do? Fire you for looking?
1 time my manager over heard me talking to a recruiter. He later met with me and asked me why I was looking & I said low pay at current place and no raise. He immediate got me a 10 K raise so it worked out for me!
The dynamics of "power" are to be experienced everywhere. While you did not deliberately negotiate a raise, your employer on the other hand deliberately WAITED till he saw you could exercise your power of flight.
Magnificent.
I would typically take the initial HR call in my car at lunch (drive to a nearby parking lot if you're worried about your current company seeing you). You could try to do this with tech screens as well, but I would make sure you have a rock solid internet connection and a good laptop. I never felt comfortable with that, so I just booked the morning off and did it at home.