My Husband and I speak different languages.
My kids upon birth, were "immersed" in both languages.
From birth, we merely spoke to them, in each of our respective languages.
My kids are now 5 and 9.
They are, fluent in both languages. In terms of speaking it and reading it and hearing it.
They are also... multi-lingual, and learn other foreign languages, via their school. And they even teach ME, about the other languages.
Now, we always have, great conversations.
In both languages.
Learning a language... should not be how you are experiencing it, with your kids.
Or, you get them a Tutor.
You say you are fluent... but not native speaking.
Still, you are fluent.
You need to have a way, to teach your kids.
Actually speaking it to them and giving them lessons on it and reading to them books in that language and watching videos in that language etc.
OR, join a culture group... for that language.
Anyone can join a culture group...and families.
That way, you interact with those that speak the language.
Your kids too.
OR send them back to their language school.
OR, you can take lessons... too.
OR, you can send them back to immersion classes. And you take classes too... and you all join a cultural group in that language.... and then you will learn the language/the culture/ and make other friends who speak the language.
Thereby, maximizing your and your kids, exposure to it all and learning it.
The thing is as well: you say you are fluent. BUT, if you don't speak it correctly and have the wrong 'accent' while speaking the language... your kids will learn incorrect enunciation and accents.
My Husband for example, even if someone is fluent in his language.... he can tell where they are from/their region or even if it is an American accent or not.