Some information about <b>on * dime</b> from ngram by books.google.com:

This construction <b>'on one's dime'</b> is more rare than the same with articles (a, the):

[![NGRAM from books.google.com][1]][1]
[1]: https://i.sstatic.net/2dZ7X.png

But what about meaning?

Here examples from books I did find:

> If the group doesn’t agree, it’s on my dime. <br><br>
> It was dinky, the landlord refused to paint it (so I did, on my dime), the appliances were old, sucked and broke down a lot and my bathroom suite was mustard yellow. <br><br>
> Katz didn't seem the type to ignore me, especially since the trip was on her dime. <br><br>
> The coffee I’m willing to get but not on my dime.”<br><br>
> Then brought her up here while you're still clocked in officially on my dime?"<br><br>
> Then “get your ass back here” became “fine, but this is on your dime, Adams.” <br><br>
> “I know that, but you were working Charles’s land, taking risks on his dime. <br><br>
> “Just don’t forget,” Jimmy coos over the phone as we discuss the details of his upcoming trip to Tokyo on my dime. <br><br>

I don't know exactly meaning of this construction but I suppose this is like "at my expense", when you care of something on your own responsibility.

I hope this will be helpful for you.