Puzzle of the Week #194 - Double Birthdays

This conversation took place on a particular day of the (non-leap) year:


“Isn’t it curious,” remarked Izzie, “that if you take today’s date, double the date number and double the month number, you get my birthday?”

“Interesting!” replied Leila, “but if you consider today as the ‘n’th day of the year and work out what the ‘2n’th day of the year is, you get MY birthday!”


Izzie’s and Leila’s birthdays are in the same calendar month.


When did this conversation take place and when are the girls’ birthdays?