I suspect there would be oxygen saturation sensors (or whatever they're called) that would cause the heart to circulate blood faster when the oxygen level drops or the body needs more, such as during exercise or excitement. Otherwise, the person would faint any time they exerted themselves beyond a certain level. I would totally believe that if I read it in a novel, even if it's not a real thing now.
Regarding your comment about a real heart transplant recipient's personality changing to resemble the donor, okay, I'm open to that idea but the scientist in me would look to the psychological effect of a major event (such as a heart transplant and the accompanying medical and emotional issues leading up to it) rather than the presence of a "thinking" organ. Comparing that to the person's loss of their own thinking organ and its replacement by a machine . . . hmmm. I think in some stories it would make a fascinating and deeply moving situation, and in others I think it would seem out of place.
As with anything in writing, it depends on what you want to say with the story and if this is the best way to show it.