How careful do you have to be when job hunting?

When looking around for new jobs, how careful do you need to be about reaching out to people with mutual connections to people at your firm on LinkedIn, applying to jobs at firms that are potentially competitors, etc.? 

I am probably overthinking this...but I have never switched jobs before (still on my first job out of undergrad) and am curious if people ever "tattle". IE - I reach out to someone at a firm I am interested in about a job for that firm, they happen to know someone at my firm, and then they tell them that I was looking around? I generally avoid reaching out to people with a lot of mutual connections at my firm for this reason. But for example, I sent an invite/message to a VP at a firm that's hiring and said I was interested in the role, yada yada. He denied my connection/message and I'm thinking oh no what if he tells so and so that I reached out...

Another hypothetical - I interview at a firm like BX/Carlyle/Oaktree, who invests as an LP on occasion with my firm. Will they potentially go tell my firm that I was seeking a role there? 

I feel like RE is a small world...am I overthinking this? Do people talk about this stuff? I would hope that it would be assumed to be kept confidential but you never know, I guess. Just because I wouldn't say anything doesn't mean everyone would.