Anyone can buy as debt (bad or otherwise) - it's a matter of negotiation between the buyer and the seller of the debt.
However, the new owner should not be making themself out to be actng on behalf of the old owner - that seems to amount to misrepresentation. Also, the old owner should send a notification letter to the debtor saying the debt has been sold and details of the buyer.
Are you sure it's really a solicitor you are dealing with? Quite a lot of firms that buy debts have in house solicitor firms and use their letterhead to send letters to debtors (presumably on the basis the debtor will be more concerned by a letter from a solicitor) but if they are contacted on the phone it is the firm's normal call centre that answers!