what does "personally involved with"here mean?
“If Landon is innocent, he has nothing to hide.”
She paused, looked Duncan in the eye. “Seriously? I doubt you’re that naive, Mr. Duncan. I know I’m not.”
“I’m authorized to compensate for information,” he began as they went up the steps into the little church proper.
“You’re authorized to pay for gossip? No, thanks. When I gossip, I do it for free.” She walked out and turned toward the parking lot and her car.
“Are you personally involved with Landon?” Duncan called out.
She felt her jaw tighten, cursed the fact he’d ruined her post-yoga mood. She tossed her mats, her bag in the car, opened the door. And in a wordless reply to his question, shot up her middle finger before she got in, turned the key and drove off. 】

does"personally involved with"=dating with sb.?

Question

what does "personally involved with"here mean?
“If Landon is innocent, he has nothing to hide.” 
She paused, looked Duncan in the eye. “Seriously? I doubt you’re that naive, Mr. Duncan. I know I’m not.” 
“I’m authorized to compensate for information,” he began as they went up the steps into the little church proper. 
“You’re authorized to pay for gossip? No, thanks. When I gossip, I do it for free.” She walked out and turned toward the parking lot and her car. 
“Are you personally involved with Landon?” Duncan called out. 
She felt her jaw tighten, cursed the fact he’d ruined her post-yoga mood. She tossed her mats, her bag in the car, opened the door. And in a wordless reply to his question, shot up her middle finger before she got in, turned the key and drove off. 】

does"personally involved with"=dating with sb.?

Answer

Sorry, I disagree with with akVince.  "to be personally involved" in something like a project means to be directly involved rather than indirectly involved.  So, "to be personally involved" in a hiring decision means that you directly participated in the decision to hire the person.

However, "to be personally involved with someone" usually means a dating/romantic/sexual relationship.

That's why the woman shot a middle finger at Duncan.  He was making a very personal insinuation which she did not appreciate at all.

Answer

From what you provided, personally involved with means some sort of relationship between She and Landon.

It's not always a dating relationship. It's possible the relationship can be about business or something like that.