Answer by Professional Japanese Teacher
Yes, どちらへ？ dochirae literally means “where are you going?”
However, it doesn’t mean that your neighbour is keeping tabs on you. It is just a greeting to acknowledge the fact that you’re going out. It is similar in English where you see your neighbour and you talk about the weather to break the silence.
In this case, you do not really have to report your whereabouts to the asker. Just reply ちょっとそこまで chotto soko made (just there) will suffice. If you want to be specific, you can say where you’re actually going.
For example, if you are going to the post office, you can say ちょっと郵便局(ゆうびんきょく)まで chotto yuubinkyoku made (just to the post office).