After an implant placement I have sinus-mouth connection, should I be worried about it?

You have a complication you need to take care of immediately. If you do not take care of it soon, it can get more serious that what you think. There should never be any communication, from sinus cavity to the mouth. Please visit an oral surgeon, because all that area needs to be cleaned, bone grafted, and closed (If the area is currently infected, the dental implant must be removed). After that you need to make it heal, at least 8 months needs to pass before you think about doing a next step. Hope this helps.

You might have had sinus infection/ cyst that drained once it got perforated All depends on the size of sinus perforation. Sometimes the membrane heals on it's own.   You should return to your surgeon.  Let them evaluate it. He / she will know exactly what happened during the surgery and fhe prognosis. Antibiotics do work for dental procedures.  There are some specific to sinus surgery / bacteria that you should be on.  

Sounds like the implant is already infected and needs to be removed and the bone repaired. Did you have a CBCT scan and a 3D printed guide made before surgery? You should never have an implant placed without guided surgery to avoid these dangerous complications.