For the most part, I don't tell people about it unless they need to know. The first time that someone different was going on, I was really young. My mother noticed that I tended to get sick every time anyone in my family got sick even if it wasn't contagious (knots my back when she had them, a tendency to get a fever only when someone in my family had one that ended almost immediately after theirs ended, etc). She worked in a hospital and, although she didn't quite believe in this stuff, she acknowledge that some people may be more sensitive to that sort of thing than others since she had seen it in the loved ones of some of her patients as well (e.g. sympathy pains husbands get while their wife is giving birth). For the most part, this was acknowledged in a way that made it clear to me that this was not something I should talk about and was only mentioned in terms of possibility of careers choices ("You should be a doctor since you seem to be good at healing," with that last part said in an embarrassed whisper). As for the emotional aspect, I do not think she wants to know; I do not think anyone really wants to know that anyone wants to know that their child remember being a toddler and remembers feeling the pain and heartbreak that comes from one parent cheating on the other and violence in the house and all of that but in the nuanced way adults involved feel it, not in the more simple emotions of a small child or remembers the feeling of completely loosing control due to rage even when never having lost that kind of control herself. The rest of my family would just think I was crazy and probably recommend I go on anti-psychotics.
To get back to your original question, if I am around people often enough to start having physical symptoms when they are sick, I eventually have to talk to them about that particular thing (but not about the rest of it) because I have it enough under control that I can generally avoid the symptoms if they avoid certain things. For the most part, if they are good friends, they eventually think I am eccentric but they love me anyway, and, if not, they think I am insane and distance themselves. I haven't had any problems with someone threatening to tell others, which would actually be a problem in my career field, but I am careful of who I tell.
There hasn't been as big of a problem with claircognizance because a lot of people who believe in a higher power can accept wording like, "It just really feels like I need to do this thing" as the voice of God leading you in the direction He wants you to go in (I grew up around Christianity so I am most familiar with the protestant Christian wording of this phenomenon).
There are some people that I have told, but I usually only tell people from cultures who are more accepting of this sort of thing. They generally think I am lucky and are happy to learn about it.
However, I find that not telling people leaves a huge gulf and if the person becomes important enough that it is better to risk a bad reaction than to continue to have that gulf between you, then it is probably time to risk it