Marissa and Melissa both want to know the other's birthday. Each of them knows her own birthday. They have the following conversation.

Marissa:What day of the month were you born on?
Melissa says what day of the month she was born on.
Marissa:I now know that our birthdays are never on the same day of the week in the same year.
Marissa says what day of the month she was born on.
Melissa:That is a prime number. However, I cannot figure out what your birth month is.
Marissa:My birth month has the same number of days as your birth month.
Melissa:The sum of the digits in the day of the month you were born on equals the position of my birth month in the year.

What is Marissa's birthday? What is Melissa's birthday?