This has not been answered anywhere else, googled extensively. Only things I found were a few "How to trade" pages, but they're not even in great detail other than some instructions in an answer by Robotnik here:
How do I trade with friends over the internet in Pokemon Sun/Moon?
However, I seem to have an issue beyond these instructions. I, and my friend, have followed these instructions to the letter, but cannot see each other in the Festival. We have been in each others NDS's as friends for years and have had no issues adding each other in previous Pokemon games. He has traded with friends in Sun/Moon, albeit locally, but I have yet to due to this issue.
Note: I do see random "passersby" but not NDS-registered friends.
Best Answer
If your guest list is full, your friend will automatically be kicked out (disappear). The only way to fix this seems to be adding them to VIP lists which guarantees they'll be there every time you enter Festival Plaza.
However, this assumes that you're already able to see them - to make them appear for the first time you'll just have to refresh your guest list until they appear. To force a re-sync of the guest list, I'd recommend both parties to disconnect and reconnect to the Festival Plaza.
Source: Reddit & personal experience