  In this article we consider the simultaneous recovery of bulk and boundary
potentials in (degenerate) elliptic equations modelling (degenerate) conducting
media with inaccessible boundaries. This connects local and nonlocal Calder\'on
type problems. We prove two main results on these type of problems: On the one
hand, we derive simultaneous bulk and boundary Runge approximation results.
Building on these, we deduce uniqueness for localized bulk and boundary
potentials. On the other hand, we construct a family of CGO solutions
associated with the corresponding equations. These allow us to deduce
uniqueness results for arbitrary bounded, not necessarily localized bulk and
boundary potentials. The CGO solutions are constructed by duality to a new
Carleman estimate.
