Notes app - How to synchronize note with other users

Not if you're talking about two separate single iCloud accounts...I'm pretty sure.

For basic note sharing via the Notes app the easiest thing I can think of is to add a new account...the same account...to both devices. Best that it's NOT another iCloud account...I don't even think that would work. An Outlook.com account, for instance, works well with iOS notes but wouldn't have all the Apple-specific features recently added to notes.

You could also share Calendars and Reminders via that account if you so wish. Otherwise you'd just turn on "Notes" and leave everything else turned off.