I think I would give different responses to the original question, depending on what is being discussed: Push Hands, sparring, self-defense, fighting, and sport all require somewhat different approaches. If we are talking about sparring as a training activity, then there should be little thought to who wins or loses, the idea is to learn and to teach cooperatively.
Without movement or the intent to move, there can be no sticking; however, I think that distinquishing full and empty is a higher principle. In sparring, first try to exploit the opponent's full and empty while hiding your own. If your opponent has his full and empty hidden, provoke him to reveal it.
If your sparring opponent is adamantly committed to stillness, then walk around him and kick him gently in the behind. If he is not quite that committed to stillness, he will move to thwart your clever plan and then you will have something to stick to.
Another strategy would be to launch a slow strike, while attempting to hide your full and empty. If there is no response, you follow through. If your opponent responds, you now have an opportunity to practice counters to the response, and your opponent can practice counters to your counters. The cycle can, in theory, continue without end if both partners have enough skill and are evenly matched.
A last approach is to recall the sayings in the Art of War that no fixed defense can succeed. If your opponent maximizes his response to right and left, then front and back are vulnerable. If he maximizes defense of right, then the left is exposed. If he defends everywhere, then everywhere is exposed.
When you spar, see how your opponent deploys his "forces." Unless he is in the Preparation Posture or matching your movements, he has already tended to expose vulnerabilities that you can probe. If he matches your every move, recall that perfect symmetry is impossible, even if you are twins.
The game of Go exhibits many of the characteristics that we talk about in Taijiquan, like full and empty. One of its peculiarities is that a beginner can attempt a strategy of exactly matching every move of this opponent and theoretically force a draw. In reality, this strategy is a losing one. The board appears perfectly symmetrical (19 X 19 intersecting lines, with play on the intersections in the interior, sides, and corners), but it is not. The players move consecutively, and there is only one center point. If you take no initiative, you will always lose to any player of skill.
The issue should not be whether you take the initiative, but how you take it. Take the initiative by giving your partner what he wants and making him choke on it.
If he wants to be still, let him be still and use it.