I can speak to athletics/track and field, the sport I sometimes cover.
Most events at a professional level and NCAA events at the national level require athletes to pass through a "mixed zone" after competition. This is (usually) a divided space with athletes on one side of a barrier and reporters on the other. This is where post-competition interviews happen.
At major international events (e.g. World Championships, Olympics) there are distinct sections of the mixed zone for TV, radio, and written media. This often means an athlete like Usain Bolt will have done multiple TV and radio interviews before arriving at the written press mixed zone. In cases like this the athlete is often removed to a press conference room to answer questions in a larger setting.
In smaller meets, it's up to the reporter to buttonhole the athlete they want to speak to, although there is often a meet media manager on hand to make sure reporters speak to the athletes they want. There are no formal requirements of the athletes, although talking to the press is an unwritten aspect of professionalism and good agents/sponsors will encourage their athletes to do so. The athletes, who have just completed a sometimes-grueling competition and are sometimes disappointed in their performance, can be unmotivated to talk and will escape if they can. On the other hand, there are classy athletes who respect the role of the press and will stand and talk as long as there are questions, regardless of their disappointment and/or fatigue. And winners are almost always easy to talk to.
I've almost never run in to an athlete in this sport whose etiquette while speaking to the press is questionable, but that's not to say it doesn't happen.