"We need nineteen women and nineteen men."
Translation:אנחנו צריכים תשע עשרה נשים ותשעה עשר גברים.
No, you can't separate the numbers. "Nineteen" is one number and its gender always agrees with the gender of the noun.
Not sure what "the opposite gender" is, but if you have the ה at the end in mind, note that the masculine numbers usually have the ה , not the feminine. So, therefore תשעה עשר is masculine and תשע עשרה is feminine.
Well, numbers are a very special case. They are still confusing at times for me. But if you'd like you can check this post about Misleading gender of words in Hebrew and numbers are also discussed and all of the numbers are listed in full. Hope it helps you:
I am constantly confused about when to add a Het at the end of the number or the עשר when used in a sentence. I suspect it has to do with the gender, but sometimes it is on the number, other times on the tenth portion, and I can't seem to figure out the rule. Is there a standard rule for this?