The sentence as quoted looks inconsistent because the first pair of nouns is in the singular (banker/customer) while the second one is in the plural (debtors/creditors). Surfing the Internet, I've found that the original text presents both pairs of nouns in the singular form:
When anyone opens a current account at a bank, he is lending the bank
money, repayment of which he may demand at any time, either in cash or
by drawing a cheque in favour of another person. Primarily, the
banker-customer relationship is that of debtor and creditor — who is
which depending on whether the customer’s account is in credit or is
overdrawn. But, in addition to that basically simple concept, the
bank and its customer owe a large number of obligations to one
another. Many of these obligations can give rise to problems and
complications but a bank customer, unlike, say, a buyer of goods,
cannot complain that the law is loaded against him." (GORDON BARRIE
and AUBREY L. DLAMOND The Consumer Society and the Law)
What appears after the em dash does not sound like good English. As OP says, there should be a tensed verb:
- Primarily, the banker-customer relationship is that of debtor and creditor — who is which will depend on whether the customer’s account is in credit or is overdrawn.
For the V-ing to work with an absolute construction, there should be a comma and the subject should be different, for example:
- Primarily, the banker-customer relationship is that of debtor and creditor, the person being one or the other depending on whether the customer’s account is in credit or is overdrawn.
In the revision above, the subject of the absolute clause can be "the person" and the non-finite "being", with "depending" being more like a preposition. In this respect, we can read on page 127 of The American Heritage Guide to Contemporary Usage and Style:
Some participles, such as concerning, considering, failing, and granting, function
as prepositions and can be used to introduce a sentence without causing a dangling
modifier. A few participial phrases, such as speaking of and judging by, also work this
Concerning the proposal, there was little debate among the board members.
Considering his reputation for honesty, his arrest came as a shock.
Speaking of exceptional performances, did you see her latest movie?
Judging by the applause, the play was a success.
Another possible parsing consists of considering "the person being one or the other" as the subject, in which case "depending" is the non-finite verb of the absolute clause.