Whether or not they express an interest in competition, my fundamental approach is to help them develop the mindset of a competitor. If we can achieve this then goals will follow. They will generally arise organically.
Some of the changes to mindset I try and cultivate in my students include:
-embrace the whole lifestyle of an athlete. Train hard then recover hard both through rest, food, water, and active recovery methods. Takeaway point: success requires an all-day approach.
-constantly strive to better yourself in your lifting. Strive for PRs whether one rep max or multiple rep maxes. Compete with yourself day in and day out. This will also tie into the development of work ethic. Compete with yourself in recovery as well as lifting.
-friendly and supportive rivalry between teammates.
-educate yourself. Study powerlifting, weightlifting, or whatever it is that interests you. Find some heroes.
-encourage attendance at meets as a spectator or volunteer or as support for your competitive lifters. Post upcoming meets on the wall or whiteboard. Make it clear that if they are interested your student can enter their first meet for the experience. I would suggest not putting a lot of pressure as far as changing the training program (specializing), worrying about hitting a certain weight class, or focusing on a total. Committing to a meet, especially if the athlete makes the decision themselves, will radically change the feel of training without the coach needing to impose such a change.
Take the right steps in cultivating the mindset of a competitor and you will see more often than not that your students will start setting their own meaningful goals.