In general - the trainer - the trainer is the one knowledgeable about proper form, proper build up, etc. The trainer gets a degree and license in the subject and therefore is the one qualified to determine the correct regiment for the athlete.
The trainer's job is to push them - but to push them to a point where there would be no injury. Their job is to make the athlete better and stronger, not to injure them.
Now in the case where an athlete hides an injury - it is on them meaning the athlete as s/he did not provide all the necessary information to the trainer. Typically (having daughters who both went on to play at the college level) - the athlete provides a full and detailed list of their medical history including any injuries and when they had them. It only helps the trainer to provide specifics so as not to re-injure and also what areas an athlete may need work on more - for example it was very important for the college trainer to know my daughter had an ACL tear and surgery - they know then what to watch for. This was specifically for basketball, however, both also played softball.
As far as a trainer knowing by seeing - they may but they may not - they may think you are slacking off that particular day, or it may be the type of injury that is harder to detect. The most important thing is honest and open communication if both player and trainer want to develop an athlete for the long term and not for say one particular competition.