Is routing necessary for person A?

Yes, in packet switched network, it is necessary to show person A what route to follow at each node (router), in the network. Home and office networks are different and far away from each other: we need some kind of inter-networking to allow them communicate.

No, routing is not necessary if we use LL leased line: modem, v.24, 35 or X.21 interface type to connect person A to B. Other switched circuits technologies such as: Frame Relay (FR), X.21, ATM, P1024B or C, would serve the same purpose. More advanced technologies would help also avoid routing: tunnel, vpn, vrf, xconnect, vpls, pseudo wire, mp-bgp, mpls labelling, etc.