I might be in the minority on this, but if I'm treating a patient, he or she doesn't have to call me doctor (although it depends on the setting). However, in the same respect of having a clinical docotorate, if I call a physician or chiropractor or podiatrist "doctor (name)" then I expect to be addressed with the same clinical respect. If they ask to be called by their first name, than I will as well. It's not an ego thing as much as a respect thing. DPT's are earning their clinical doctoral degree to treat patients as Doctor of Physical Therapy much like Doctors of Podiatry or Doctors of Chiropractic or Doctors of Optometry. The examination and treatment process is much like a physicians. Patients are getting care for the neuromusculoskeletal and/or cardipulmonary systems from an advanced and specialized clinician.
With that said, I'm not sure if it would be appropriate to introduce myself "Hi I'm Dr. Freddy. I'm here to walk you to the bathroom and back to bed today." On the other hand, if it comes to a low back patient, I can manage low back pain with manual therapy, therapeutic exercise, modalities, and refer to an MD for prescription. In that sense, if a patient wants to think of me as their doctor, I'm quite alright with that. I'll be the one primarily treating them and building a relationship for the patient to learn to self-manage. And if it helps them to thinik that the "doctor" told them to do such and such exercise, then I'll gladly use the title.
I hope that makes sense and isn't too scattered of a thought.