Connotation of 行きますか vs 行きましょうか

Question

My textbook has the following conversation:

A: 明日どこに行きますか。

B: デパートに行きましょう。

A: どのデパートに行きますか。

B: 貴方の家の北のデパートに行きましょう

The third line is translated to English in the textbook as "which department store should we go to?" My trouble is with the connotation of this sentence.

I think that the 〜ます stem seems to imply a certainty expected upon B, that is, that B already knows where we're going. In English I would expect this more to be "which department store are we going to?" The "should we go to" seems to me to ask for suggestions (rather than a unilateral decision by B). If that connotation is what is desired, then I think that the third line would be better written as どのデパートに行きましょうか。

Am I wrong about the connotation of this sentence, or am I right in thinking that the 〜ます stem is implying a stronger level of certainty from B than expected?

Answer 1

You're correct in thinking that "B" is not giving a direct certainty in the form of "We will go to X". However, in this context, the suggestion is a more explicit suggestion than "Should we go to X" as if it were a question. It's probably closer to "Let's go to X" as opposed to "We will go to X". So, applying that level of suggestion, let's translate the example:

A: 明日どこに行きますか。
A: Where are we going tomorrow?

B: デパートに行きましょう。
B: Let's go to a department store.

A: どのデパートに行きますか。
A: Which department store will we go to?

B: 貴方の家の北のデパートに行きましょう
B: Let's go to the department store north of your house.

Answer 2

Judging from the given context, I think that there is a difference in initiative in the conversation between "A" and "B". Clearly "B" has the initiative. "A" simply asks for "B"'s intention so "A" uses ～ます stem. On the other hand, "B" states his own judgment.
～しょう stem used by "B" is an expression for acting with partner "A". Therefor I think that the translation starting with "Let's" in psosuna's answer is correct.
If "B" intends to act alone, I think that "B" will use "～ます" instead of "～しょう" like the following conversation.

A: 明日どこに行きますか。
B: デパートに（ひとりで）行きます。
A: どのデパートに行きますか。
B: 貴方の家の北のデパートに（ひとりで）行きます。