Most dentists - for most procedures - aim for a painless experience. If there is reason to keep some pain sensation intact, the dentist will inform you, and ask at appropriate intervals if you can feel pain.
The efficacy of lidocaine and other local anesthetics depends on how closely your nerve distribution comes to the norm (they will inject the environs of the "normal" anatomical position of the nerve), how much anesthetic is injected, whether there are local factors which alter the local tissue pH (e.g. presence of an abscess or infection), how quickly it is removed from the site, etc.
Is this feedback that the dentist can use to detect issues with the actual work as he goes along, or would the only purpose of reporting it be to try to get him to do something to mitigate it, for the sake of my immediate experience?
Local anesthetics prevent pain sensation, but not vibration or pressure, which themselves can be unpleasant. Hypothetically, a dental procedure which should be painless with anesthesia can be carried out equally well whether the patient feels pain or not, so if you are stoic, feeling pain and not telling your dentist will not likely affect the outcome of that procedure. However, informing the dentist helps them to know if your anatomy is different (valuable information for the next time they need to do a similar local or regional block, your response to the anesthetic used (they might try another anesthetic or approach in the future), or if there might be an unseen infection altering the pH (and efficacy) of the anesthetic. Therefore it is probably wise to let your dentist know what you're feeling. They can reassess the situation, then offer you relief of some kind: a painless injection (the second injection into an already partially anesthetized region is truly painless), nitrous oxide (a gas), or other.
I would liken this for most instances to suturing a laceration. There is never a time when I wish the patient to feel pain while I'm addressing the repair. I address what I see. If a patient feels pain, I want to know that and address the situation.
An Update on Local Anesthetics in Dentistry