Some information about on * dime from ngram by books.google.com:

This construction 'on one's dime' 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.

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.

Katz didn't seem the type to ignore me, especially since the trip was on her dime.

The coffee I’m willing to get but not on my dime.”

Then brought her up here while you're still clocked in officially on my dime?"

Then “get your ass back here” became “fine, but this is on your dime, Adams.”

“I know that, but you were working Charles’s land, taking risks on his dime.

“Just don’t forget,” Jimmy coos over the phone as we discuss the details of his upcoming trip to Tokyo on my dime.

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.

About style maybe somebody can say more but it's look like for me much less formal than "at my expense".

  I hope this will be helpful for you.

Some information about on * dime from ngram by books.google.com:

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

But what about meaning?

Here examples from books I did find:

If the group doesn’t agree, it’s on my dime.

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.

Katz didn't seem the type to ignore me, especially since the trip was on her dime.

The coffee I’m willing to get but not on my dime.”

Then brought her up here while you're still clocked in officially on my dime?"

Then “get your ass back here” became “fine, but this is on your dime, Adams.”

“I know that, but you were working Charles’s land, taking risks on his dime.

“Just don’t forget,” Jimmy coos over the phone as we discuss the details of his upcoming trip to Tokyo on my dime.

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.

About style maybe somebody can say more but it's look like for me much less formal than "at my expense". And that it's appear not so long time ago I think it can't be formal anywhere. I hope this will be helpful for you.

Some information about on * dime from ngram by books.google.com:

This construction 'on one's dime' 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.

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.

Katz didn't seem the type to ignore me, especially since the trip was on her dime.

The coffee I’m willing to get but not on my dime.”

Then brought her up here while you're still clocked in officially on my dime?"

Then “get your ass back here” became “fine, but this is on your dime, Adams.”

“I know that, but you were working Charles’s land, taking risks on his dime.

“Just don’t forget,” Jimmy coos over the phone as we discuss the details of his upcoming trip to Tokyo on my dime.

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.

Some information about on * dime from ngram by books.google.com:

This construction 'on one's dime' 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.

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.

Katz didn't seem the type to ignore me, especially since the trip was on her dime.

The coffee I’m willing to get but not on my dime.”

Then brought her up here while you're still clocked in officially on my dime?"

Then “get your ass back here” became “fine, but this is on your dime, Adams.”

“I know that, but you were working Charles’s land, taking risks on his dime.

“Just don’t forget,” Jimmy coos over the phone as we discuss the details of his upcoming trip to Tokyo on my dime.

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.

About style maybe somebody can say more but it's look like for me much less formal than "at my expense".

I hope this will be helpful for you.