I was running into the same problem trying to share notes with my wife for a number of weeks now. I think it started in early to mid October, 2019. I had shared a number of notes with her in the past, usually by text, and suddenly she could not get them to work. It would go through the process of retrieving the note, but when she tried to open it, she would get the message:
“This note can’t be opened right now. Please try again later.”
I did all kinds of things including closing out all apps, shutting down phones, shutting off my phone (which wasn’t as current of an iOS as my iPad) and even logging her out of her Apple ID and logging back in.
None of that worked.
However, I discovered that when I added her iCloud email address (which she never uses and does not wish to use) to her contact info on my devices that I could email her a shared note invitation through that unused iCloud email address. It still did not allow me to send her shared notes via text, but it worked.
Another workaround I discovered was to take an old shared note that we were no longer using and just completely change the text to be the new shared note. That worked fine as well. I also emailed a couple of blank note templates that, when I choose to change them, will already be available in her Notes.
The problem did not pose itself if she wished to share a note with me. That is why I believe that the issue was because the notes were not going through an iCloud email address. I am keeping my fingers crossed that the system will sort itself out after her iCloud address is in my contacts for a while, so that Messages can also send her shared notes.
I do not know what stupid glitch is causing the problem, but I suspect it has to do with recent updates. Perhaps Apple while sort this out for us in the near future.