Identifying changes in contact during contact-rich manipulation can detect task state or errors, enabling improved robustness and autonomy. The ability to detect contact is affected by the mechatronic design of the robot, especially its physical compliance. Established methods can design physical compliance for many aspects of contact performance (e.g. peak contact force, motion/force control bandwidth), but are based on time-invariant dynamic models. A change in contact mode is a discrete change in coupled robot-environment dynamics, not easily considered in existing design methods. Towards designing robots which can robustly detect changes in contact mode online, this paper investigates how mechatronic design can improve contact estimation, with a focus on the impact of the location and degree of compliance. A design metric of information gain is proposed which measures how much position/force measurements reduce uncertainty in the contact mode estimate. This information gain is developed for fully- and partially-observed systems, as partial observability can arise from joint flexibility in the robot or environmental inertia. Hardware experiments with various compliant setups validate that information gain predicts the speed and certainty with which contact is detected in (i) monitoring of contact-rich assembly and (ii) collision detection.