I don't have a cough but sometimes when I go into a place a moment later a person starts to cough. Last time I met my friend that I didn't seen for a long time, having some conversation after a few minutes he suddenly coughs. It keeps bothering me, what does it mean? When I go into a place or talk to someone they will start coughing.
No, it just happens and you're overthinking about it. It means absolutely nothing.
Cough is a protective reflex, which helps to clear the large breathing passages from fluids, irritants, foreign particles and microbes. A cough can also be psychogenic or neurogenic but certainly not telepathic or magic.
What causes coughs:
Most coughs are caused by cold or flu.
Other causes include: smoking, allergies(for example hay fever), infections like bronchitis, mucus dripping down the throat from the back of the nose. A cough is very rarely a sign of something serious like lung cancer.
So, the question is, if cough is contagious in a similar way as yawning is, only with this difference that you don't even cough?
Yes, such things can happen. It's not exactly that you "spread" the cough, but you can--unintentionally-- "provoke" the person to cough. This can happen if the person feels uncomfortable in your presence, because you are, for example, giving awkward, ambivalent messages (when your words, voice, emotions, gestures and facial expressions seem to be in a conflict). The other person can find this scary or funny, but he/she is not able or does not want to tell you, so the tension builds up in his/her chest/throat resulting in an urge to cough.
The problem can be either on your or the coughing person's side.
Speaking from experience; been on both sides.