We use shared notes for shopping. For a long time getting them in sync on two iPhones, or iPhone and Mac, have been completely unreliable. Sometimes they sync, mostly not. This used to be reliable. Anybody seeing this ?
Yes. Messages too.
I got Notes originating on my Mac to appear on my iPad by force-quitting Notes on the iPad, turning off Notes sync in iPad’s iCloud prefs, waiting 2 minutes, then turning on Notes sync.
Similar steps failed to achieve Messages sync.
I have two iMacs on my LAN, as well as an iPhone. One iMac is my personal work machine, while the other hosts my web server. After reading your post, out of curiosity, I went over to my server machine and launched the Notes app. At first there was nothing except for a new note placeholder. However, within minutes it synced with my personal work machine Notes app. Every note that shows on my work machine also appears on my server machine, so no issues here.
Occasionally, I have seen the non-syncing issue in the Messages app, but that is between my work machine iMac and my iPhone Xs Max. Sometimes I think it may have something to do with Internet congestion, or perhaps Apple’s servers are just being overtaxed. Who knows.
Maybe not the response you expected or will find useful, but…
My wife and I have been using AnyList for managing our (shared) shopping list for a few years now, and it’s hands-down the best app in that category that I’ve encountered thus far.
AnyList is built by a very small and committed indie team, is updated regularly, and is very reasonably priced. Well, OK, unlike Notes it’s not entirely free; but IMHO it’s well worth its moderate price tag.
P.S.: “Just a happy customer,” as they say when a disclaimer might be in order. 
