Pulsatile Tinnitus

It's by no means uncommon. As we age, and as our carotid artery (runs just by the inner ear) becomes narrowed by deposition of plaque, the blood being forced through the narrow parts is going to cause vibrations that the inner ear detects. It's the same phenomenon as a hearing aid that uses bone for transmission of sound through to the cochlea. Further, if we have age/diet-related deposition of fat, the visceral kind (not the spare tire outer kind), that acts as a transmitter of the heart's powerful contractions, and those 'waves' travel throughout the body. Head position, the type of pillow, they matter as well.

What specialist do I see for this? Cardiologist, ENT?

An audiologist might be able to confirm what you hear, and let you know why, but also a nutritionist if you need to alter your diet. A cardiologist would soon know if you have hypertension and need to be placed on something like metoprolol to calm your heart so that it doesn't beat so strongly. Or, a sleep coach/expert who knows of this problem and how intrusive it can be, and may have some techniques or tips on how to deal with it effectively.

I'm just guessing at this since I don't have much of that problem (sometimes yes, most of the time no) and have never investigated it and asked for expert help.