Numerous answers to your question could be given. Far too many for a short-answer format such as this, in fact. I'll give one answer.
Browning's use of a first-person speaker in both poems allows characterization to be revealed about those speakers. The speakers indirectly condemn themselves, at least from a reader's point of view.
In "Duchess," the speaker's pomposity, arrogance, self-importance, and egocentricity are revealed by the speaker's words. When he lists people and actions and animals and bits of nature (imagery) that brought joy to his deceased wife, and lists these as the behaviors she exhibited that led him to order her to be killed, he reveals his true nature and personality. This allows the reader to "discover" the Duke's personality and character, and draw his/her own conclusions. The dramatic monologue form, including the silent listener (a representative of the father of the Duke's new fiance who, in effect, is being threatened by the Duke), provides the occasion for the Duke to indirectly reveal himself. The listener's silence, of course, also reflects the Duke's personality--he's not one to let others talk. His ideas are the only ones that count.
In "Lover," the first-person speaker reveals his personality (characterization) in much the same way. Again, the speaker is unaware of the reaction a reader will have to his narrative. He describes a moment of beauty and purity. As he explains the moment of Porphyria's true and pure love, and how he captures it and maintains it, he inadvertently reveals his warped and egocentric mind. Porphyria brings to the scene peace and calm and love and purity, and the speaker sees his act as in line with peace and calm and love and purity. She dies, he is sure, without pain. Pain would destroy the moment, so he refuses to accept or acknowledge it. He reveals his deranged mind.
First-person speakers allow Browning to reveal characterization indirectly through unaware narrators. First-person speakers allow Browning to let the speakers condemn themselves, and to let readers discover that for themselves.