I think secrecy was a holdover from my parent's generation. My own mum could not discuss birth control with me the week before my wedding! I simply asked what she used, not about sex and she said to talk to my doctor. She was born in the 1920s and I gather times were different!
I remember when my roommate who was an actress, was asked to do a voiceover for a Tampax commercial and she refused because she did not want it on her resumé! That was probably the mid 70s.
We never told our boyfriend or husband unless we had to. It was not polite conversation, so I understand exactly where your parents were coming from -- the dark ages where women's bodies were dirty and scary. Even among women it was "the curse", "my friend" and so on... Ridiculous!
I think those Tampax ads helped grow us up. There is nothing dirty or scary or weird about females, or the biology of a female body.
With our daughter we discussed it years in advance by and through the study of biology. Why did our dog get neutered? How do cats get pregnant? Why does that mummy have a big tummy? Why is Willow Mum buying sanitary pads and tampons? Later, how did a baby get in there? How is it fed or breathing in there? The questions came naturally. The answers did too.
I'd make sure that your partner knows the biology as well as you do. Many of us think we know more than we do. (I have a menstrual cycle -- of course I know all about it.) So I'd suggest you both study enough to know the answers and then be prepared to answer as honestly and openly as you can. Use the teachable moments as they show up. Do not put a great question off until later.
When my daughter at 5 asked why I was buying tampons in the grocery store, I said something like, "You know I menstruate, right? These tampons help me stay comfortable, like a diaper keeps a baby comfortable." She did not ask more, but when she did, I answered. She had as much right to that information as any human does. Later, when she saw tampons in my bathroom and noticed they did not look like diapers, I showed her with a doll, where they went and explained it did not hurt. Later she told her dad that she had a hole in her vulva and asked where his hole was. It's hard not to laugh or be embarrassed but he was a trouper. He said he was a man and that instead of a vagina, uterus and vulva, he had a penis and testicles. Out came the big book we had about our bodies and he showed her. It was the first time she heard about ova and sperm and all that, but she was fascinated by it. Her best question had come years earlier when she saw her biological father in the shower, "Daddy what is that thing coming out of your vulva?"
Overtime we added to the information. We made sure there were books and that she was free to ask anything. Sure there are things we did not answer. "How often do you and Dad have sex?" Our answer was that it was private between partners but that sex was a part of most love relationships and that different people enjoy sex more or less than other people.
When we noticed that she was growing hair and had little breast buds forming, we bought her pads and I showed her again how they stuck to her undies. She chose to wear pantyliners and because of that her first cycle was no big deal. She had a pad in her purse and was wearing a liner. I asked if she wanted to celebrate becoming a woman and she chose a RED party at a pizza joint with some of her friends -- boys and girls.
It was not a big moment for her, but she liked the party. Since then she's come to me with a problem about pimples on her bum and to her dad about a pimple on her vulva -- she was staying at his house that evening. If we don't have an answer, lucky us -- the world of knowledge is as close as our phones.