The hinged region at B is not a support, actually. Thats just a connection (or to be more precise, a pinned connection). A pinned support is something different; a pinned support is something what you see on the right end C. The support itself is connected to the ground (for example), and cannot translate in any of the directions (which means that the right end of beam at C cannot translate in any direction at all). However, the hinged connection at B can translate anywhere it wants (because it is not connected to the ground, so the ground is not providing any reaction force in the vertical or horizontal direction over there at all). The hinge at B means that both the beams have to translate together in vertical and horizontal axis, but can rotate freely there (with respect to each other), so it means that they will transfer the forces but no bending moment.
When you apply a load P at the direction shown in the first figure, the right beam is trying to move down, and hence hinge at B is supposed to move down (since there is no attachment to ground at that location, which will resist it from moving down). When B moves down, it is asking the left beam also to move down (because that hinge at B includes the right end of left beam pinnedly connected to the left end of right beam). This means that both supports, i.e. A and C wants to move down but cannot because of the supports applied at them (since those are connected to the grounds, which basically provides a reaction force by resisting their down motion). This is the reason why you will see stresses in both the beams, and not just the right beam.
So, now lets see the force reactions at each. Isolate the right beam, do statics and you will find P/2 as the force on the point C and on the left end of right beam (which is hinged with the left beam). This force is transferred to the other left beam, and at the reaction A, this shear force should be resisted.