For "collapse" to occur, the interaction of the quantum system must be with a classical system. A classical system being something big, noisy that can be well approximated by classical mechanics.
Simple interaction between two systems doesn't cause "collapse" it makes the two systems become entangled. Classical systems are a bit contagious in this sense, anything that gets entangled with them becomes classical.
To be a bit more precise, this distinction between classical and quantum is a bit our fault. Everything is quantum at a fundamental level, classical system is one for which we do have not have a precise knowledge of the state of the system, instead we have a coarse representation. This should make more obvious in which way "classicalness" is contagious. Since the knowledge of a part was coarse, the knowledge of the newly entangled system is also necessarily coarse.
