For best clarity, and when you’re client in MP game. Look for the mission log to see if there is note under mission info, stating: “Only the host will have the mission progress saved.” (link to the image in my 2nd reply).
If there is no such note in the mission info, you, as a client, have all prerequisites completed and mission saves for you as well. However, if you do see that note, something is missing and mission progress won’t save for you.
I know that it is a complex system and i have no idea why devs made it so.
Also, when i 1st played GZ, i did it solo. And after i completed all main mission and almost all side missions, i then joined MP with my 2nd, 3rd and 4th character, to level them up by doing missions with the host. Though, i’ve seen that “note” as well, when host picked up side mission that i didn’t find in my solo game beforehand. We completed it together and when i went back to my solo game, i had to locate, start and complete that side mission again, for it to be saved for me as well.