With the official release of 1.3.27 I have been doing some experimentation on my current 2 player Farm. I will add more information as results come in, but in terms of what I have gathered so far...
Price Raising Skills from levels (eg. Artisan): Shared
When items are dropped in the shipping box, it applies the bonus as long as at least one person on the server has Artisan (+40% value to Artisan Goods) regardless of who drops it into the box.
Here I (with Artisan) created and dropped into the shipping box the Apple Jelly, while the host (with Agriculturalist) created and dropped in the Orange Jelly. Both has the +40% value applied. The Shipping box we used was the one that comes with the farm.
Price Raising Items: Individual
When I obtained the Bear's Knowledge, my Blackberries and Salmonberries had the 3x price raise applied but berries sold by my Host partner without still sold for berries for regular price (even if I was the one who picked the berries).
Once getting Botanist, only the forage that I pick-up is Iridium Quality. Forage picked-up by my host partner has the regular spread of quality.
The rate at which the crops grow is dependent on who plants them. In the images below the Crops on the left are planted by The Host (with Agriculturist) and the crops on the right are planted by me (Artisan). All Cauliflower were planted on
Day 1 of Spring.
Day 9 Harvest(Agriculturist plus Deluxe Speed-Gro):
Day 10 Harvest(Artisan plus Deluxe Speed-Gro):
Day 11 Harvest(Agriculturist):
Day 13 Harvest(Artisan):
I am not sure how having multiple professions will stack, but I plan on testing this once we get a bit more cash and xp on our farm.
Once we both hit max level in the different skills, and have lots of cash floating around, testing should be easy with the Re-spec statue.