Hello Anon,
As the poem is a dramatic monologue, we have to assume that the tone of the poem is the tone of the speaker, i.e. of the duke.
The duke's apparent self-revelation is only puzzling, if we think of him as a real person. Since this is a dramatic monologue, the "silent auditor" and the reader are one: thus if Browning is to reveal certain aspects of his imaginary character to the reader, he must also reveal them to his imaginary auditor.
The trick lies in providing the reader with so much detail, that if the reader does happen to wonder "Why is the duke telling the envoy all this?", he will himself be able to confabulate a convincing reason for the apparent indiscretion.
MrP